Akram ABBASPOUR*, Davood FARSADIZADEH, Mohammad Ali GHORBANI

Department of Water Engineering, University of Tabriz, Tabriz 51666-14766, Iran

Estimation of hydraulic jump on corrugated bed using artificial neural networks and genetic programming

Akram ABBASPOUR*, Davood FARSADIZADEH, Mohammad Ali GHORBANI

Department of Water Engineering, University of Tabriz, Tabriz 51666-14766, Iran

Artificial neural networks (ANNs) and genetic programming (GP) have recently been used for the estimation of hydraulic data. In this study, they were used as alternative tools to estimate the characteristics of hydraulic jumps, such as the free surface location and energy dissipation. The dimensionless hydraulic parameters, including jump depth, jump length, and energy dissipation, were determined as functions of the Froude number and the height and length of corrugations. The estimations of the ANN and GP models were found to be in good agreement with the measured data. The results of the ANN model were compared with those of the GP model, showing that the proposed ANN models are much more accurate than the GP models.

artificial neural networks; genetic programming; corrugated bed; Froude number; hydraulic jump

1 Introduction

The transition of a supercritical open channel flow into a subcritical flow is associated with the formation of a hydraulic jump. Hydraulic jumps have been extensively studied because of their frequent occurrence in nature and their use as energy dissipators in outlet works of hydraulic structures (Hager 1992).

A complete description of a hydraulic jump also involves its length Lj, which is the distance between the two cross-sections with the sequent depth y2and upstream supercritical depth y1. From the practical point of view, the jump length is an important variable to define the downstream limit beyond which no bed protection is necessary. The jump length is hard todefine in actual experiments, mainly because the end cross-section of the hydraulic jump is difficult to locate due to surface waves and residual turbulence (Hager 1992).

Because of the complexity of hydraulic jumps, more practical tools are required to model hydraulic jump processes. Regressions have been most commonly used to estimate jump characteristics. However, regression analysis may have large uncertainties, and the computed jump depth and length can be far from the actual ones. Also, the regression analysis has some limitations caused by predefined equations for modeling.

Recently, artificial neural networks (ANNs) and genetic programming (GP) have been used to model hydraulic jump processes. They have been used to estimate the scouring around piles by Kambekar and Deo (2003) and the scouring below spillways by Azmathullah et al. (2008). Also, a combination of the fuzzy inference system (FIS) with ANNs, ANFIS, has been employed to estimate the wave characteristics by Mahjoobi et al. (2008). GP and ANNs have been successfully applied in maritime engineering (Kalra and Deo 2007; Singh et al. 2007; Gaur and Deo 2008).

The purpose of this study was to investigate the characteristics of hydraulic jumps in a horizontal flume with a corrugated bed using the ANN and GP methods. These soft computing tools can evaluate the relative importance of input parameters, such as the relative roughness, the corrugation wavelength, and the Froude number, on the jump process.

2 Materials and methods

2.1 Experimental setup

The experimental setup consisted of a main flume in a discharge collection channel. The main flume was 0.25 m wide and 0.50 m deep, and had a bed slope of 0.002. A triangular weir was placed at the end of the channel to measure the discharge. A supercritical approach flowwas produced using a sluice gate. A corrugated polyethylene sheet with sinusoidal corrugations of wavelengthsand heighttwas installed perpendicular to the flow direction in the flume so that the corrugation crests were at the level of the upstream bed carrying the supercritical flow. The flow channel section of the experiment is illustrated in Fig. 1 (Abbaspour et al. 2009). A total of 123 experimental groups were conducted. Ranges of the variables in the experiment are shown in Table 1. Hydraulic jumps on the corrugated bed were produced for different Froude numbers, and the hydraulic parameters were measured. The water surface profiles of the jumps on the corrugated bed were measured at the centerline of the flume with a point gauge with an accuracy of 0.1 mm. The supercritical depthy1and sequent depthy2of the jumps were continuously measured using ultra sonic sensors, and the data was saved on a computer and processed with the VisiDAQ software. The length of the jump,Lj, in the experiment was recorded. The values of the Reynolds number in this experiment were in the range of 61 200 to 175 600.

Fig. 1Sketch of free jump on corrugated bed in experiment (Abbaspour et al. 2009)

Table 1Ranges of field data in experiment

whereELis the difference between the specific energy before and after the jump, andEL=E2?E1.

2.2 Artificial neural network (ANN)

An artificial neural network (ANN) is an information processing paradigm that is inspiredby the way biological nervous systems, such as the brain, process information. It is composed of a large number of highly interconnected processing elements (neurons) working in unison to solve specific problems. Neurons are arranged in layers, including an input layer, hidden layers, and an output layer. There is no specific rule that dictates the number of hidden layers. The function is established largely based on the connections between the elements of the network. In the input layer, each neuron is designated for one of the input parameters. The network learns by applying the back-propagation algorithm, which compares the neural network simulated values with the actual values and calculates the estimation errors. The data set in the network is divided into a learning data set, which is used to train the network, and a validation data set, which is used to test the network performance. In the present study, the neural network fitting tool (nftool) of MATLAB 7.5 was used.

After training the network, verification is conducted until the success of the training can be established. In the simulation of hydraulic jumps, characteristic data were investigated with the neural network using the Levenberg-Marquardt algorithm, which is an approximation of Newton’s method. In order to check the sensitivity of the neural networks, a simulation study was carried out with hidden nodes of different numbers, 5, 10, 15, and 20.

The correlation coefficient (R), the root mean square error (RMSE), the mean absolute error (MAE), and the Nash-Sutcliffe efficiency coefficient (NSE) statistics were used to evaluate the model accuracy. R shows the degree to which two variables were linearly related. Different types of information about the predictive capabilities of the model are measured through RMSE and MAE. An efficiency of 1 (NSE = 1) corresponds to a perfect match of the modeled values to the observed data.

whereXiis the observed values,Xis the mean ofXi,Yiis the estimated values,Yis the mean ofYi, andnis the number of data sets.

2.3 Genetic programming (GP)

In artificial intelligence, genetic programming (GP) is an evolutionary algorithm-based methodology inspired by biological evolution to find computer programs that perform a user-defined task. GP initializes a population consisting of random members known as chromosomes, and the fi tness of each chromosome is evaluated with respect to a target value. The principle of Darwinian natural selection is used to select and reproduce fi tter programs. GP creates computer programs that consist of variables and several mathematical function sets as the solution. The function set of a system can be composed of arithmetic operations (+, ?, ×, ÷), function calls (such as ex,x, sqrt, and power), even relational operators (>, <, =) or conditional operators, and a terminal set with variables and constants (x1,x2,…,xn). An initial population is randomly created with a number of individuals formed by nodes (operators, variables, and constants) and previously defined according to the problem domain. An objective function must be defined to evaluate the fitness of each individual. Selection, crossover, and mutation operators are then applied to the best individuals and a new population is created. The whole process is repeated until the given generation number is reached (Koza 1992).

The fitness of a GP individual may be computed using Eq. (9):

whereXjis the value returned by a chromosome for the fitness casej, andYjis the expected value for the fitness casej.

In the GP model many operators, like sin, cos, and log, and mathematical functions were used, and it was found that the functions of the proposed GP model were complex. Also, the GP model using more operators has larger estimated difference. In this study, for simplicity, only four arithmetic operators (+, ?, ×, ÷) were used. The functional and operational parameter settings used in the GP model are shown in Table 2. The performance of the GP model in training and testing sets was validated in terms of the common statistical measuresR,RMSE,MAE, andNSE.

Table 2Parameters of GP Model

3 Results and discussion

3.1 Hydraulic jump estimation using ANN model

Different ANN structures were tried in terms of hidden layer node numbers. In this study, the number of neurons in the hidden layer was obtained using the trial and error method. From the simulation study, which was carried out using the ANN model, it was found that with 15 neurons in the hidden layer, the estimation accuracy increased to some extent.

Fig. 2Comparison of measured and estimatedvalues using ANN model for training, validation, and testing data

Fig. 3Comparison of measured and estimatedvalues using ANN model for training, validation, and testing data

Fig. 4Comparison of measured and estimatedvalues using ANN model for training, validation, and testing data

3.2 Hydraulic jump estimation using GP model

The superior performance of the GP model, compared with other methods, is attributed to the powerful artificial intelligence techniques for computer learning inspired by natural evolution to find an appropriate mathematical model to fit a set of points. GP employs a population of functional expressions and also numerical constants, based on how closely they fit to the corresponding data (Koza 1992).

Fig. 5Comparison of measured and estimatedvalues using GP model for training and testing data

Fig. 6Comparison of measured and estimatedvalues using GP model for training and testing data

Fig. 7Comparison of measured and estimatedvalues using GP model for training and testing data

where1C,2C,3C,4C,5C, and6Care constant coefficients that are determined by the GP model (Table 3).

Table 3Constant coefficients in GP model

3.3 Comparison of ANN model with GP model

The ANN and GP models are compared in Figs. 2 through 7. It can be seen from the fit line equations (the equations are assumed to bey=ax+b) in the scatter plots of the GP model that the coefficientsaandbfor the ANN model, with a higherRvalue, are, respectively, closer to 1 and 0 than the GP model. This can be clearly observed from its fit line equation coefficients.

Table 4 compares the ANN and GP models, with all statistical measures,R,RMSE,NSE, andMAE, of the training and testing data. According to Table 4, the ANN model has lower absolute error as compared with the GP model, showing that the proposed ANN models are much more accurate than the GP models for water engineering.

Table 4RMSE,MAE,R, andNSEstatistics of training and testing data of ANN and GP models

4 Conclusions

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(Edited by Yun-li YU)

*Corresponding author (e-mail: akabbaspour@yahoo.com)

Received Jul. 19, 2012; accepted Feb. 27, 2013