PREDICTION OF HYDRODYNAMIC
COEFFICIENTS OF PERMEABLE
PANELED BREAKWATER USING
ARTIFICIAL NEURAL NETWORKS
Mona A. Hagras
Irrigation & Hydraulics Department, Ain Shams University, Cairo, Egypt monahagras@yahoo.com
Abstract
In the present study, Artificial Neural Networks (ANNs) with different topologies have been evaluated to be used to predict hydrodynamic coefficients of permeable paneled breakwater. Two neural network models are constructed, one to predict wave transmission coefficient (Kt) and another for the prediction of wave reflection
coefficient (Kr). Back propagation algorithm was used to train a multi-layer feed-forward network (Levenberg
Marquardt algorithm). The capability of ANN topologies to estimate these coefficients is evaluated using the Mean Squared Error (MSE). Based on training patterns of different ANNs, a 5-7-1 topology has been selected to predict both coefficients. The results of the developed ANN models proved that this technique is reliable in such field. A good match between the measured and predicted values was observed with correlation values varying in the range (0.9508-0.9805) for the training set and (0.9159-0.9877) for the testing set.
Keywords: Artificial Neural Network; Wave transmission; Wave reflection; Permeable paneled breakwater 1. Introduction
The development of coastal areas depends on shore protection against waves and currents. There are many types of coastal protection structures such as artificial beaches, nourishment, breakwaters, jetties, seawalls, artificial headlands and groins. Many types of breakwaters can be used such as rubble mound breakwater, massive vertical face breakwaters as block type, caisson type, cellular type, composite breakwaters as vertical super structure of plain concrete blocks or caissons resting on a large foundation of rubble mound, and flexible breakwaters (row of contact piles or sheet piles). The previous types are fully protection breakwaters and the others are partial protection breakwaters such as floating types, submerged types, detached wall types and permeable breakwaters like screen types, slotted vertical barriers, perforated caissons, array closely pipe breakwaters, pile supported vertical wall breakwater or skirt breakwater. Permeable breakwaters are increasingly perceived as an environmental friendly design and many types of structures have already been examined [Ahmed, (2011)]. A permeable pile pipe breakwater was suggested by [Mani and Jayakumar, (1995)].The possibility of using partially submerged porous breakwater was studied by [Mani, ( 2000)]. A permeable breakwater in form of thin, rigid, vertical slotted wall made from concrete or timber planks was preliminary studied by [Lamb, (1932)] and recently by [Dalrymple and Martin, (1998)] in order to increase the wave attenuation efficiency and the breakwater stability. Permeable breakwaters have already been implemented in form of pile breakwaters, which are formed from a series of piles, placed in rows [Sundar, (2003)] and implemented in form of curtain wall-pile breakwaters at Yeoho port (Korea), [Suh et al., (2007)]. [Balah, (2006)] studied experimentally the permeable paneled breakwater as a wave dissipater. Even there is a great volume of published work dealing with permeable breakwaters, it is noticed that there is a lack of a simple mathematical model for these structures for predicting breakwater characteristics and its performance depending on the Hydrodynamic coefficients.
Artificial Neural Network (ANN) can be useful tool to predict the hydrodynamic coefficients of permeable breakwater. The stability and reliability analyses of coastal structures such as rubble mound breakwaters using neural networks have been carried out [Mase et al, (1995)]; [Kim and Park, (2005)]. Similarly [Mandal et al., (2009)] have applied neural network to predict the wave transmission of the floating breakwater. The current study aims to develop quick and easy reliable method to predict the hydrodynamic coefficients; wave transmission and wave reflection of permeable paneled breakwater using Artificial Neural Networks.
2. Theoretical Approach
1. Transmission Coefficient; Kt
Kt
(1)
2. Reflection Coefficient; Kr
K R (2)
3. Energy dissipation Coefficient; CD
CD ELE 1 K K (3)
According to the evaluation of the changes of these three coefficients, due to the change in the breakwater geometrical parameters and the surrounding wave climate, the optimum structural features of the breakwater and the best wave conditions to be working within in order to achieve the best efficiency were recommended. The objective of this study is to predict both wave transmission (Kt) and reflection (Kr) coefficients through the
suggested breakwater system shown in Figure (1). For two-dimensional regular wave tests, these two coefficients are assumed to be function of many variables, which are illustrated in Figure (2).These variables were divided into three groups, (Balah, 2006) as follows:
I. Variables concerning the motion of waves; h , HI , L , T ,
II. Variables concerning the medium where the waves are traveling; g , w , μ , θ
III. Variables concerning the geometrical configuration of the proposed breakwater; b , D , G , α , y
By applying Buckingham's theorem considering the three repeating variables as (L, g, w), then the
dimensionless π terms may be derived and the functional relation may be rewritten as:
Kt = ƒ (HI/gT2 , h/L , L/b , D/L , G/b , y/h , μ2g/L3 2 , α , , θ ) (4)
By the same way, the functional relation for the coefficient of reflection was deduced as follows:
Kr = ƒ (HI/gT2 , h/L , L/b , D/L , G/b , y/h , μ2g/L3 2 , α , , θ ) (5)
Whereas the tested incident waves are perpendicular to the breakwater model, then ( ) is considered constant as well as the bed slope (θ) which is kept constant. Therefore these variables are excluded. Similarly the wall inclination angle (α), as all the tested models were vertical. The term (μ2g/L3 2) is excluded, as viscous force may be neglected. Finally, the functional relationships describing the transmission coefficient Kt and similarly
the reflection coefficient Kr may be derived as
Kt = ƒ (HI/gT2 , HI /y, h/L , L/b , G/D , G/b , y/h) (6)
Fig. 1. The considered geometry and wave parameters
Fig. 2. Different views of the tested model
3. Experimental Setup
a rotating steel hand in the back of the flume. The flume is also provided with a pump, reservoir and a measuring discharge device to produce currents with wide range of velocity. The layout and the details of the flume are shown in Figure (3).Two wave absorbers were built, one at each flume end to prevent reflection of waves from its both ends or reduce it to acceptable levels. A variable speed flap type wave generator was manufactured to produce regular waves with different periods and heights.
Fig. 3. Layout and details of the wave flume
4. Artificial Neural Networks
An artificial neural network is a type of biologically inspired computational model, which is based loosely on the functioning of the human brain. It is more useful to think of a neural network as performing an input-output mapping via a series of simple processing nodes or neurons [Alvisi, ( 2005)].
In this study the most common neural network type, the multilayer perceptron, was adopted. This type of network is formed by three or more layers of basic computing units named artificial neurons or nodes. It includes an input layer, an output layer and a number of hidden layers with a certain number of active neurons connected by feed forward links, to which are associated modifiable weights. In addition, there are also bias, which are connected to neurons in the hidden and output layers. The number of nodes in the input layer denotes the number of independent variables and the number of nodes in the output layer stands for the number of dependent variables (Haykin, 2008). To obtain the best prediction of the output parameters different network structures and architectures were elaborated and evaluated. The optimum number of hidden layers and the optimum number of nodes in each of these cases was found by trial and error. Mean Square Error (MSE) is used in the current study to compare the performances of regression models.
In training phase, the back-propagation supervised learning technique has been used for training the neural networks. It is most useful for feed forward networks. The back propagation needs toknow the correct output for any input parameters. The input parameters that influence the wave transmission (Kt) and wave reflection
(Kr) of permeable paneled breakwater such as HI/gT2, y/h, G/D , Kh , G/b are considered. The main objective
of Back propagation Neural Network (BNN) technique is to train the network such that the result outputs are nearer to the desired values.
BNN has two phases in training. Phase one is concerned as the forward stage, where the information is propagated from the input to the output layer. Second phase is the backward stage where an error, defined as the discrepancy between the observed value and the desired nominal value in the output layer, is propagated backwards in order to adjust the weightings and bias values. In the forward phase, the weighted sum of input components, uj, is calculated as:
n
j ij i j
i 1
u
w x
bias
(8)
where wij denotes the weight between the jth and the ith neurons in the preceding layer, xi stands for the output
of the ith neuron in the preceding layer, and biasj alludes to the weight between the jth neuron and the bias
j j
y
f u
(9)Where, f refers to the activation function. In the current study, the sigmoid activation function as expressed in Eq. (10) was used
T y 1
(10)
To ensure statistical significance of the attained results, in each simulation, the available data were randomly divided into two mutually exclusive partitions: the training set, with two-thirds of the available data, used during the modeling phase, and the test set, with the remaining one-third of the examples used after training in order to evaluate the model performance [Souza et al., (2002)]. The overall objective of training algorithm is to reduce the MSE defined as,
MSE ∑ S S^ (11)
where Si is the actual value, S^i is the predicted value, and n is the total number of data records [Vicente et al.,
(2012)
5. Results and Discussion
MATLAB NN Tool Box was used to construct two Artificial Neural Networks structures for prediction of hydrodynamic coefficients; wave transmission (Kt) and wave reflection (Kr) for permeable paneled breakwater.
MATLAB Tool Box selected the maximum training Epochs in optimum way to avoid over training phenomena which is the most typical problem when Neural Network based models are created. Figures (4) & (5) show the performance of proposed neural network models for both Kr and Kt respectively during training and test phases.
Fig. 5. Performance of Proposed ANN for Kt prediction
The correlation coefficient is calculated to know how best the network predicted Kt and Kr values match the
measured Kt and Kr values respectively. The straight line is drawn at an angle of 45o between the two axes to fit
the data points. A high correlation is obtained when all the points lies exactly on this straight line.
The network predicted wave transmission (Kt) and wave reflection (Kr) are calculated using Eq. (14) & (15)
respectively. At the end of each training process Correlation Coefficient (CC) is calculated between measured (desired) Kt and predicted Kt as same as Kr using following equation
CC ∑∑ ∑
(12)
Where
x = x-x/
Xx = network predicted Kt values or Kr values
x/ = mean of x
y = y-y/
y = measured Kt values or Kr values
y/ = mean of y
After training and testing of both network models, CCs are calculated between desired output and network output using Eq. (12).
In the present study, updated algorithms such as Levenberg-Marquardt algorithm (LM) is used to train the two network models with 21 epochs for wave transmission (Kt) prediction and 36 epochs for wave reflection (Kr)
prediction. The best trained, validated and tested ANN models correlation coefficients (CC) and Mean Square Error (MSE) of Kt and Kr are shown in Tables (1) & (2) respectively. The final trained and tested results CC of
two network models are shown in Figures (6) & (7).
Table 1. Correlation Coefficient (CC) and Mean Square Error (MSE) for best ANN topology for (Kt) prediction
Hidden Nodes
CC Training
CC Validation
CC Testing
MSE Training
MSE Validation
MSE Testing
Epochs
ANN1 5 0.9751 0.9749 0.9793 1.09e-3 9.79e-4 9.76e-4 29
ANN2 6 0.9776 0.9723 0.9737 9.48e-4 1.24e-3 1.37e-3 17
ANN3 7 0.9805 0.9810 0.9877 7.97e-4 8.39e-4 7.23e-4 36
Table 2. Correlation Coefficient (CC) and Mean Square Error (MSE) for best ANN topology for (Kr) prediction
Hidden Nodes
CC Training
CC Validation
CC Testing
MSE Training
MSE Validation
MSE Testing
Epochs
ANN1 5 0.9389 0.8918 0.9233 1.11e-4 1.71e-4 1.40e-4 22
ANN2 6 0.9384 0.9256 0.8781 1.17e-4 1.37e-4 1.80e-4 14
ANN3 7 0.9508 0.9339 0.9159 8.59 e-5 1.49e-4 1.40e-4 21
ANN4 8 0.9485 0.9441 0.9133 9.16 e-5 1.46e-4 1.41e-4 20
It is observed that the correlation coefficients obtained are above 0.90. A high correlation coefficients (CC training =0.9805, CC validation=0.9810, CC testing=0.9877) are obtained at epoch equal to 21 with hidden nodes equal to 7 for ANN model of Kt prediction. Also a high correlation coefficients (CC training =0.9508, CC
validation=0.9339, CC testing=0.9159) are obtained but at epoch equal to 36 with hidden nodes equal to 7 for ANN model of Kr prediction.
Fig. 6. Correlation between prediction and measured Kt for ANN3 with 5-7-1 topology
0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Kt Pr
edicted
V
a
lues
Kr Measured Values
CC train = 0.9805
Fig. 7. Correlation between prediction and measured Kr for ANN3 with 5-7-1 topology
After training the network model, weights and biases of each network are fixed. These fixed weight and bias values are shown in Figures (9) & (10) for Kt and Kr respectively. Each input value gets multiplied with the
weight and adds with bias value. The total sum is the input at each hidden node and pass through a transfer function as defined in Eq. (10), and further the output from hidden node get multiplied with the weight and adds with the bias value and total sum pass through sigmoid function again as shown in Eq. (10).
The wave transmission (Kt) and wave dissipation (Kr) is estimated using following formulations:
Transfer Function F N 1
,
i = 1 to 7 (13)Where, Ni are values of hidden nodes and Fi are the transfer functions of hidden node i.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.160 0.180
Kr
Pr
edicted
V
a
lues
Kr Measured Values
CC Train = 0.9508
HI/gT2 y/h G/D kh L/b Kt
Input Layer Hidden Layer Output Layer
bias bias -2.477 -1.228 1.250 0.7 68 0.5 52 -2 .42 2 3.8 30 -0.014 -0.229 -0.011 -0.337 0.244 0.3 60 0. 168 0.09 1 -2.1 42 -0.332 2.248
-0.39-0 4 .2 88 -0.8 12 -1.3 92 -2.5 49 2.6 78
-0.857
-0.521 -1.643 -1.540 0.9 84 -0.1 78 0.260 -0.836 -0.008 -1.523 1.8 02 +1.151 +1.7 55 -0 .72 4 +0 .74 8 -0 .2 2 6 -1 .7 4 2 +5 .0 1 6 -0 .9 4 9 3 .70 7 -0.3 52 2.310 0.188 0.882 -0.1 10 0.2 39
Fig. 10. The ANN structure of Kr prediction with weights and biases
For Kt ANN model, the trained hidden nodes and its transfer functions are
N1 = HI/GT2 (-1.228) + y/h (-0.0143) + G/D (0.0913) + kh (0.984) + L/b (-1.392) + 1.151
N2 = HI/GT2 (-2.477) + y/h (-0.229) + G/D (-2.142) + kh (-0.179) + L/b (-2.549) + 1.755
N3 = HI/GT2 (1.250) + y/h (-0.012) + G/D (-0.332) + kh (0.260) + L/b (2.678) - 0.725
N4 = HI/GT2 (0.768) + y/h (-0.338) + G/D (2.249) + kh (-0.836) + L/b (-0.857 + 0.748
N5 = HI/GT2 (0.552) + y/h (0.245) + G/D (-0.394) + kh (-0.009) + L/b (-0.521) - 0.227
N6 = HI/GT2 (-2.423) + y/h (0.361) + G/D (-0.289) + kh (-1.523) + L/b (-1.644) -1.743
N7 = HI/GT2 (3.830) + y/h (0.168) + G/D (-0.812) + kh (1.803) + L/b (-1.540) +5.0164
F1 = (2 / (1+exp (-2N1))) -1
F2 = (2 / (1+exp (-2N2))) -1
F3 = (2 / (1+exp (-2N3))) -1
F4 = (2 / (1+exp (-2N4))) -1
F5 = (2 / (1+exp (-2N5))) -1
F6 = (2 / (1+exp (-2N6))) -1
F7 = (2 / (1+exp (-2N7))) -1
N1 to N7 and F1 to F7 represent summation function and transfer function at each hidden node respectively and
Kt = F1(3.708)+ F2(-0.353)+ F3(2.310)+ F4(0.188)+ F5 (0.882)+ F6(-0.110)+ F7(-0.239)- 0.949 (14)
Eq. (14) provides trained ANN model for estimating wave transmission (Kt) of permeable panel breakwaters.
For Kr ANN model, the trained hidden nodes and its transfer functions are
N1 = HI/GT2 (-0.637) + y/h (0.948) + G/D (-3.917) + kh (0.790) + L/b (-1.622) - 4.899
N2 = HI/GT2 (0.352) + y/h (0.282) + G/D (1.566) + kh (-0.692) + L/b (0.844) -0.789
N3 = HI/GT2 (1.344) + y/h (0.957) + G/D (0.966) + kh (-0.116) + L/b (1.618) -0.498
N4 = HI/GT2 (-1.742) + y/h (-0.134) + G/D (0.094) + kh (0.235) + L/b (0.202) -0.575
N5 = HI/GT2 (0.0322) + y/h (1.521) + G/D (2.426) + kh (0.436) + L/b (-3.147) -3.454
N6 = HI/GT2 (2.487) + y/h (1.095) + G/D (-0.530) + kh (-0.609) + L/b (2.023) + 2.312
N7 = HI/GT2 (-2.153) + y/h (-1.117) + G/D (0.414) + kh (0.388) + L/b (-2.340) -2.344
F1 = (2 / (1+exp (-2N1))) -1
F2 = (2 / (1+exp (-2N2))) -1
F3 = (2 / (1+exp (-2N3))) -1
F4 = (2 / (1+exp (-2N4))) -1
F5 = (2 / (1+exp (-2N5))) -1
F6 = (2 / (1+exp (-2N6))) -1
F7 = (2 / (1+exp (-2N7))) -1
N1 to N7 and F1 to F7 represent summation function and transfer function at each hidden node respectively and
Kr is computed as.
Kr = F1(0.221) + F2(-0.174) + F3(0.173) + F4(-0.537) + F5(0.134) + F6(-0.627) + F7(-0.709) -0.026 (15)
Eq. (15) provides trained ANN model for estimating wave transmission (Kr) of permeable panel breakwaters.
6. Conclusion
The encouraging results obtained in this work show that the ANNs can be very useful as tools to predict hydrodynamic coefficients of permeable paneled breakwater. In this study diverse ANN architectures were developed and evaluated to predict hydrodynamic coefficients; wave transmission coefficient (Kt) and wave
reflection coefficient (Kr) of permeable paneled breakwater using the Mean Squared Error (MSE). Both ANNs
selected to predict both coefficients have a 5-7-1 topology each. A good match between the measured and predicted values was observed with correlation values varying in the range of 0.9508 to 0.9805 for the training set and in the range of 0.9159 to 0.9877 for the testing set. Two equations have been obtained to predict the hydrodynamic coefficients one each.
7. LIST OF SYMBOLS
b : Length of panels [L] C : Wave celerity [LT-1]
CD : Wave energy dissipation coefficient [--]
D : Distance between panels' vertical axes [L] EI : Incident wave energy [ML2T-2]
EL : Wave energy losses [ML2T-2]
ER : Reflected wave energy [ML2T-2]
ET : Transmitted wave energy [ML2T-2]
g : Gravitational acceleration [LT-2] G : Space between panels [L] h : Water depth [L]
hf : Free board of panels [L]
HI : Incident wave height [L]
HR : Reflected wave height [L]
HT : Transmitted wave height [L]
K : wave number [L-1]
Kt : Wave transmission coefficient [--]
L : Wave length [L] T : Thickness of panels [L] T : Wave period [T]
Y : Draft depth of submerged panels [L]
α : Panels vertical inclination [--]
: Angle of incident wave to the breakwater [-]
γw : Water unit weight [ML-2T-2] μ : Water dynamic viscosity [ML-1T-1]
θ : Bed longitudinal slope [--]
References
[1] Ahmed, H. (2011): Wave Interaction with Vertical Slotted Walls as a Permeable Breakwater. PhD thesis, Institute for Geo-technic,
Waste Management and Hydro Sciences (IGAW), University of Wuppertal, Germany.
[2] Alvisi, S. G. Mascellani, M. Franchini, and A. B´ardossy, (2005): Water level forecasting through fuzzy logic and artificial neural
network approaches, Hydrol. Earth Sys. Sci. Discuss., 2, 1107–1145.
[3] Balah, A. (2006):The use of permeable paneled breakwater as a wave energy dissipater. Master thesis. Ainshams University. Irrigation
and hydraulics department. Cairo. Egypt.
[4] Dalrymple, R.A., and Martin, P.A., (1990): Wave diffraction through offshore breakwater. Journal of Waterway, Port, Coastal, and
Ocean Engineering, ASCE, Vol. 6, No. 6.
[5] Haykin, S. (2008): Neural Networks and Learning Machines. Prentice Hall, New Jersey, USA.
[6] Kim, D H and Park, W S. (2005): Neural network for design and reliability of analysis of rubble mound breakwaters. Ocean
Engineering- Elsevier Journal. 32, 1332-1349.
[7] Lamb, H., (1932): Hydrodynamics. 6th edition. Cambridge University Press. 738 P.
[8] Mandal S., S.G. Patil and A V Hegde, (2009): Wave transmission prediction of multilayer floating breakwater using neural network.
Proceedings of International Conference in Ocean Engineering, ICOE .
[9] Mani, J.S., (2000): Porous breakwater for sea defense, a new concept. ICOPMAS Conference, ASCE, Iran.
[10] Mani, J.S., and Jayakumar, S., (1995): Wave transmission by suspended pipe breakwater Journal of Waterway, Port, Coastal, and
Ocean Engineering, ASCE, Vol. 121, No. 6, November.
[11] Mase H., Masanobu S. & Tetsuo S. (1995): Neural network for stability analysis of rubble mound breakwater. J. Waterway, Port,
Coastal and Ocean Eng, 121(6), 294-299.
[12] Souza, J., Matwin, S. & Japkowicz, N. (2002): Evaluating data mining models: a pattern language. In: Proceedings of 9th Conference
on Pattern Language of Programs. Illinois, USA.
[13] Suh, K. D., Jung, H. Y. & Pyun C. K., (2007): Wave reflection and transmission by Curtain wall-Pile breakwaters using circular piles.
J. of waterways, Port, Coastal and Ocean Eng., Vol.34, Issues 14-15, pp. 2100-2106.
[14] Sundar, V. & Subbarao, B. V. V., (2003): Hydrodynamic performance characteristics of quadrant front face pile supported breakwater.
J. Waterways, Port, Coastal and Ocean Engineering, vol.129, no.1, pp.22-33.
[15] Vicente, H. et al., (2012): Prediction of the quality of public water supply using artificial neural networks. Journal of Water Supply:
