Optimal design of triangular side orifice using multi-objective optimization NSGA-II

Triangular orifices are commonly used in various industrial and engineering applications, including fluid metering, flow control, and measurement. Discharge prediction of triangular orifices is an essential area of research in fluid dynamics. The discharge coefficient of a triangular orifice is a function of the orifice geometry and the flow conditions. Accurately predicting discharge is crucial for ensuring the proper functioning of these systems and can be used to optimize the design of triangular orifices. Conventional methods for discharge prediction, such as using empirical equations, have limitations in terms of accuracy and applicability. One approach to predicting discharge for triangular orifices uses computational fluid dynamics (CFD) simulations. CFD simulations use mathematical models to solve the equations of fluid motion and predict fluid flow behavior. These simulations can provide detailed information about fluid flow characteristics, such as velocity and pressure distributions, in complex geometries. However, CFD simulations can be computationally intensive and require significant computational resources. Another approach to predicting discharge for triangular orifices is using artificial neural networks (ANNs). ANNs are machine learning algorithms that are designed to model complex relationships between inputs and outputs. In recent years, ANNs have been applied to predicting discharge for orifices with promising results, even in cases where the relationships between the input parameters and the discharge are complex.

Numerous studies examined the hydraulic behavior of side orifices in open channels with various shapes. Regression equations were proposed for spatially varied flow analysis for the square side orifice (Gill 1987; Hussain et al. 2011) and flow characteristics of rectangular side orifices (Ramamurthy et al. 1986, 1987) and rectangular side sluice gates (Swamee et al. 1993; Ghodsian 2003; Esmailzadeh et al. 2015) and circular side orifices under different flow conditions (Hussain et al. 2010, 2016; Vatankhah & Bijankhan 2013; Guo & Stitt 2017). Additionally, artificial intelligence tools, namely group method of data handling, ANN, adaptive neuro-fuzzy inference system and genetic algorithm were employed to predict the discharge coefficient of rectangular, square and circular side orifices (Ebtehaj et al. 2015; Eghbalzadeh et al. 2016; Azimi et al. 2017; Noman et al. 2022; Ayaz et al. 2023).

Qian et al. (2019) generated 26 models using different combinations of input parameters and identified the most effective model among them. The model that utilized input variables W/H, B/L, H/y₁, and F_r1 successfully predicted the coefficient of discharge (C_d) with an root mean squared error (RMSE) of 0.0122 and an R-value of 0.9598.

Recently, regularized extreme learning machine (RELM) models were proposed to predict the discharge coefficient of triangular side orifices and compared with the extreme learning machine (ELM) model. Researchers suggested that the RELM model (R = 0.995 and RMSE = 0.003) showed higher efficiency than the ELM model (R = 0.982 and RMSE = 0.010) (Mahmoudian et al. 2022; Moghadam et al. 2022). Using artificial intelligence, Shen et al. (2022) predicted the discharge coefficient (C_d) of triangular side orifices. They employed a back-propagation neural network (BPNN) to train the model and used a sparrow search algorithm (SSA) to optimize its weights and thresholds. They found that the SSA-BPNN model had high accuracy and strong generalization ability, with a maximum error of 6.56% and an average error of 1.73%. Sensitivity analysis using Sobol's method identified W/H, F_r1, and B/L as primary factors in optimizing the discharge capacity of the side orifices. Gerami Moghadam et al. (2022) concluded that the generalized structure group method of data handling (GSGMDH) outperformed the classical group method of data handling (GMDH) in predicting the coefficient of discharge (C_d) of triangular side orifices and suggested that the most effective parameters were the upstream Froude number and orifice height ratio.

Based on the above discussion, extensive literature on the discharge estimation for circular, rectangular, and square orifices is available. Recently, discharge through a triangular orifice has been observed experimentally, and machine learning techniques have also been employed to predict discharge. However, to the authors' knowledge, no literature is available for predicting discharge through triangular orifice using ANN. Moreover, no study is available for the most efficient triangular orifice, i.e., the geometrical properties of a triangular orifice for the maximum discharge passing through it. Hence, in the present study, ANN models were obtained to predict the discharge through the triangular orifice and velocity downstream of the main channel. The developed ANN models were used to obtain the most efficient triangular orifice section simultaneously, ensuring the maximum velocity downstream of the main channel using a novel hybrid multi-objective optimization model (NSGA-II) that uses genetic algorithms. A detailed comparative analysis of the ANN model proposed in this study with the earlier models proposed by researchers is presented.

Vatankhah & Mirnia (2018) performed experiments in a laboratory flume to gather data on the discharge and the relevant hydraulic parameters. They tested a range of orifice sizes viz. orifice length (L), orifice height (H), orifice crest height (W) and discharge through the triangular side orifice (Q_s), and recorded the corresponding upstream head (y₁), upstream velocity (V₁), and downstream head (y₂). The experiments were conducted in a hydraulic laboratory, using a 12 m long, 0.25 m wide, and 0.5 m deep horizontal rectangular channel connected to an upstream supply through a stilling tank, and a tailgate to regulate flow depth. The equilateral triangular side orifices made up of 0.01 m thick Plexiglas sheets with a crest thickness of about 1 mm, and the downstream edge beveled to a 45° angle were installed in the wall of the main channel, and their discharge characteristics were measured for 12 different geometric configurations (six orifices, each with two crest heights). The study collected a total of 570 runs of experimental data, including flow discharges and flow depths measured using piezometer taps and a point gauge with an accuracy of 0.1 mm. Triangular and rectangular flow measuring weirs were calibrated using an electromagnetic flowmeter with a precision level of ±0.5% of the full scale. The results showed that the discharge passing through the side orifice spilled into a diversion channel and then to the rectangular return channel, and the flow depth of the side channel did not affect the flow condition in all performed experiments that remained free flowing. Table 1 presents the statistical variation of experimental data for the equilateral triangular side orifices under free-flow conditions.

An artificial neural network (ANN) is a machine-learning model that is inspired by the structure and function of the human brain. It is composed of interconnected nodes, known as artificial neurons, which process and transmit information. Each neuron receives input from other neurons, processes the information, and then outputs a signal to other neurons in the network. The overall behavior of the network is determined by the connections between the neurons and the strength of the signals they transmit.

The process of training an ANN involves adjusting the weights of the connections between the neurons so that the network's output matches the desired output for a given set of inputs. This is accomplished using an optimization algorithm, such as stochastic gradient descent (SGD) or Lavenberg Marquardt (LM) algorithm that minimizes the difference between the network's predicted output and the actual output. Once trained, an ANN can be used for a variety of tasks, such as image classification, speech recognition, and natural language processing. The performance of an ANN can be improved by using deeper networks with more hidden layers, using convolutional neural networks (CNNs) for image data, or using recurrent neural networks (RNNs) for sequential data (Zurada 1994; Schalkoff 1997; Kheyroddin et al. 2018; Ebadi-Jamkhaneh & Ahmadi 2021; Ahmadi & Kioumarsi 2023; Ayaz et al. 2023).

Training a feedforward back-propagation artificial neural network (ANN) is the process of adjusting the weights of the connections between the neurons so that the network's output matches the desired output for a given set of inputs. The most common method for training a feedforward back-propagation ANN is supervised learning, in which the network is provided with a labeled training dataset and the goal is to learn the mapping from inputs to outputs.

The training process involves iteratively presenting the network with a sample from the training dataset and computing the error between the network's predicted output and the actual output. This error is then backpropagated through the network to update the weights of the connections so that the error is reduced on the next iteration. This process is repeated multiple times until the error reaches a satisfactory level or a maximum number of iterations is reached.

The weights of the connections are updated using an optimization algorithm, such as SGD, that minimizes the difference between the network's predicted output and the actual output. The optimization algorithm uses the gradients of the cost function with respect to the weights, which are computed using the back-propagation algorithm. The back-propagation algorithm uses the chain rule of calculus to propagate the error from the output layer to the input layer, computing the gradients of the cost function with respect to each weight in the network.

The training process can be influenced by several factors, such as the choice of activation function, the architecture of the network, the size of the training dataset, and the choice of optimization algorithm. In addition, overfitting can occur if the network becomes too complex and learns the noise in the training data, leading to poor performance on unseen data. To avoid overfitting, various regularization techniques, such as early stopping, dropout, and weight decay, can be used.

The Levenberg–Marquardt (LM) algorithm is a widely used optimization algorithm for training ANN and other nonlinear systems. It is a combination of the gradient descent and the Gauss–Newton methods and provides a way to efficiently find the minimum of a nonlinear cost function.

The LM algorithm starts by assuming that the cost function can be approximated by a linear function near the current solution. The gradient descent method is then used to update the parameters of the network in the direction of the steepest descent. If the approximation is good, the update should result in a reduction of the cost function. If the approximation is not good, the algorithm switches to a Gauss–Newton update, which considers the curvature of the cost function and provides a more accurate update.

The key idea behind the LM algorithm is to balance the accuracy of the linear approximation with the speed of convergence. This is achieved by using a damping factor, known as the Marquardt parameter, which is adjusted dynamically during the optimization process. If the approximation is good, the Marquardt parameter is reduced, allowing the algorithm to converge more quickly. If the approximation is not good, the Marquardt parameter is increased, providing a more accurate update.

The LM algorithm has several advantages over other optimization algorithms, such as being relatively fast and robust, and not requiring a good initialization of the parameters. It has been widely used in various fields, such as computer vision, control systems, and machine learning, and has achieved good results on a wide range of nonlinear optimization problems (Hagan & Menhaj 1994).

NSGA-II (Non-dominated Sorting Genetic Algorithm 2) is a popular multi-objective optimization algorithm used to solve problems where multiple conflicting objectives exist. It is a type of evolutionary algorithm that is based on the principles of genetic algorithms.

NSGA-II works by sorting the solutions into different levels of non-domination and preserving diversity in the population by balancing the trade-off between the conflicting objectives. The algorithm starts with an initial population of candidate solutions, which are then evaluated based on multiple objectives. The solutions are sorted into different levels of non-domination, and the best solutions are selected for the next generation through a tournament selection process.

Crossover and mutation are then performed to generate offspring, and the new solutions are added to the population. The sorting and selection process is repeated for multiple generations until a satisfactory set of non-dominated solutions is obtained.

NSGA-II is widely used in various fields such as engineering design, finance, and environmental management to address multi-objective optimization problems. It has been shown to be effective in handling large-scale, complex problems with multiple conflicting objectives.

In this study, two different feedforward back-propagation ANN models namely, ANN-1 and ANN-2, have been developed to predict the discharge through the triangular side orifice (Q_s) and the downstream velocity in the main channel (V₂), respectively. The discharge through the side triangular orifice (Q_s) and the downstream velocity (V₂) primarily depends on the length of the orifice (L), height of the orifice (H), crest height of the orifice (W), upstream flow velocity (V₁), and upstream flow depth (y₁). Both ANN-1 and ANN-2 models comprise three layers – input, hidden, and output layer. The input layer has five neurons, and the output layer has one neuron. L, H, W, V₁, and y₁ have been used as inputs corresponding to five neurons of the input layer while Q_s as the output to ANN-1 model. ANN-2 model is developed to predict the downstream velocity in the main channel (V₂). In this model, L, H, W, V₁, and y₁ have been used as inputs while V₂ as the output corresponding to neurons of input and output layers, respectively.

The MAE, AAD, MSE, and R provide only the average values of model performance parameters. To show the pointwise variation of errors, two different plots, namely, the error histogram and the stem plot have been used in this study. Figures 6 and 10 show the error histogram with 20 bins for ANN-1 and ANN-2 models, respectively. It is observed that the numbers of instances of fallen data are high for bins having lesser error ranges. Also, with the increase in the error ranges of bins, the corresponding number of instances decreases following a bell-shaped error distribution. Stem-plots between the data points and corresponding prediction errors for ANN-1 and ANN-2 models are shown in Figures 8 and 12, respectively. In these plots, it was observed that most of data points fall within the smaller error range of the order of 10⁻⁴ and 10⁻², respectively. Performance results presented above are in good agreement with experimental results, which demonstrate the capability and practical applicability of proposed ANN models in predicting the discharge through the triangular side orifice.

Past researchers Vatankhah & Mirnia (2018), Jamei et al. (2021), and Moghadam et al. (2022) calculated the discharge using Equation (4) by assuming that y₁ = y₂. However, this assumption does not validate the experimental observations where y₁ ≠ y₂. As a result, Mehraein (2019) proposed Equation (6), which is highly data dependent with 12 regression coefficients, failed to provide any improvement over this assumption. Additionally, Vatankhah & Mirnia (2019) suggested Equation (10) for computing discharge through the triangular side orifice where y₁ was used instead of H_CL using Equations (11) and (12) for calculating the C_d; however, Equation (11) did not provide a promising improvement over other equations. Equation (12) yielded good results over other equations, but it did not consider the F_r1 for discharge estimation, although it should not have been neglected, as suggested by Qian et al. (2019) and Shen et al. (2022). In the present study, the proposed ANN-1 model for the estimation of Q_s considering H, L, W, V₁, and y₁ as primary factors within the ranges shown in Table 1 can precisely simulate the discharge through the triangular side orifice with the MSE of the order of 10⁻⁸ and R ≈ 1.

The prediction of discharge through triangular orifices is essential for proper functioning and design optimization in various industrial and engineering applications. Traditional methods like empirical equations have limitations in terms of accuracy and applicability, while CFD simulations can be computationally intensive. However, ANNs offer a reliable and efficient alternative to conventional methods for predicting discharge coefficients, especially in situations where the relationships between the input parameters and discharge are complex.

In the present study, the ANN-1 model developed for predicting discharge through the triangular side orifice (Q_s) has an MAE of 0.00011 and MSE of 1.99 × 10⁻⁰⁸, with an AAD of 1.32% and R = 0.99945. Another model ANN-2 developed for predicting the downstream velocity (V₂) has an MAE = 0.00323, AAD = 0.937%, and MSE of 2.44572 × 10⁻⁰⁵ with a correlation coefficient, R = 0.99867. In addition, the most efficient triangular orifice has been identified by employing a hybrid multi-objective optimization model using GA (NSGA-II) to find optimal values of design parameters for maximizing discharge and downstream velocity simultaneously, thereby obtaining a series of optimal solutions in terms of Pareto front.

Moreover, a comparative study between the existing discharge equations proposed by various researchers and the ANN-1 model suggests that the developed ANN-1 model is capable of precisely simulate the discharge through the triangular side orifice with the MSE of the order of 10⁻⁸ and R ≈ 1.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.