A 3D numerical model of nearshore wave field behind submerged breakwaters

Coastal Engineering 83 (2014) 190–204

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Coastal Engineering journal homepage: www.elsevier.com/locate/coastaleng

A 3D numerical model of nearshore wave field behind submerged breakwaters Amir Sharifahmadian ⁎, Richard R. Simons Civil, Environmental and Geomatic Engineering Department, Chadwick Building, UCL, Gower Street, London WC1E 6BT, UK

a r t i c l e

i n f o

Article history: Received 20 January 2013 Received in revised form 15 October 2013 Accepted 18 October 2013 Available online 10 November 2013 Keywords: Submerged breakwater Wave field 3D wave transmission Numerical modeling Artificial neural network Radial-basis function

a b s t r a c t The functional design of submerged breakwaters is still developing, particularly with respect to modeling of the nearshore wave field behind the structure. An effective design tool needs to calculate both 2D and 3D effects. A numerical method for predicting the spatial transmission coefficient for regular waves in the shadow region of a 3D submerged breakwater is proposed in this paper. Two distinct models have been developed using machine learning algorithms; these artificial neural networks, based on multi-layer perceptron (MLP) and radial-basis function (RBF) methods, have been designed and trained against new laboratory experimental data expressed in terms of both dimensional and non-dimensional parameters. Comparisons between the experimental data and predictions from the trained models show that the non-dimensional RBF model is able to best predict the 3D wave field around the submerged breakwater. The performance of the model was validated in interpolation, extrapolation and at larger scale using different laboratory facilities, revealing sufficient agreement with the experimental results to suggest that it has potential as a design tool in real applications. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Wave-generated turbulence, surge and currents created in the surf zone as the result of wave breaking cause sediment transport and consequently shoreline erosion and accretion. Because of population growth in coastal areas as a result of urban development and recreational activities, shoreline erosion has emerged as a serious concern. Various coastal defense structures such as dikes, revetments, seawalls, and emerged or submerged breakwaters are often adopted to protect the shoreline either by stopping waves reaching the shoreline or at least forcing them to break and to some extent dissipate their energy. Among these structures, submerged breakwaters are increasingly preferred due to lower construction costs and the significant dissipation of incident wave energy in the sheltered area leeside of the breakwater in an environmentally friendly way. Retaining a clear view of the sea, preserving alongshore transport and maintaining high water quality leeside are the additional key features of submerged breakwaters. Constructing submerged breakwaters is intended to reduce erosion, trap natural sediments and restore beaches. However, shoreline response to submerged breakwaters is particularly influenced by the nearshore circulation and wave field behind the structure driven by both 2D and 3D coastal processes such as permeability through the body (2D), wave overtopping over the crest (2D) and diffraction through the heads (3D). 2D wave transmission has been widely studied. Wave transmission coefficient Kt defined as the ratio of transmitted to incident ⁎ Corresponding author. E-mail address: [email protected] (A. Sharifahmadian). 0378-3839/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.coastaleng.2013.10.016

wave height is widely used by researchers to study the wave transmitted behind coastal structures. Several different 2D design tools have been proposed to calculate Kt behind 2D submerged breakwaters: (Buccino and Calabrese, 2007; Calabrese et al., 2002; d'Angremond et al., 1996; Goda and Ahrens, 2008; Panizzo and Briganti, 2007; Seabrook and Hall, 1998; van der Meer et al., 2005). However, the only information provided by these models is the average value of Kt rather than the spatial distribution of wave height required of a reliable and accurate engineering design tool. What is required is a method of predicting wave diffraction around the head of the structure combined with the wave dissipation by overtopping. Very few studies on the 3D effects (Bellotti, 2004; Caceres et al., 2008; Hur et al., 2012; Johnson et al., 2005) and even fewer on combined diffraction and overtopping effects of similar structures have been conducted in the past (Buccino et al., 2009; Losada et al., 1996; Vicinanza et al., 2009). The majority of these studies are restricted to low-crested structures with a small number considering fully submerged breakwaters. This was the stimulus for the present study, which is intended to improve nearshore wave field modeling around submerged breakwaters with particular emphasis on flow circulation and associated sediment transport. The spatial distribution of the nearshore wave field around submerged breakwaters has been investigated numerically using datadriven algorithms called Artificial Neural Networks (hereafter ANNs). Such models have proved to be useful in many fields of engineering to solve complicated nonlinear problems where many relationships are involved. ANNs have actually been applied successfully to some coastal engineering problems (Mase et al., 1995; Medina, 1999; Panizzo and Briganti, 2007; van Gent and van den Boogard, 1998; van Gent et al., 2007; van Oosten et al., 2006).

A. Sharifahmadian, R.R. Simons / Coastal Engineering 83 (2014) 190–204

In the present study, two different ANN models have been designed and trained using new experimental data and the predicted transmission coefficients compared against each other and against actual data measured in facilities of different size. The main body of the paper is divided into six sections. The introduction offers an overview of the topic, justification for the study, and a brief literature review. Section 2 presents a brief description of artificial neural network models and the basic concepts including an explanation of the specific neural network models used in this study. A description of the experiments, the methods of data selection, and approaches for data pre-processing to prepare the training data are set out in Section 3. An investigation of existing design tools is presented in Section 4. Section 5 concerns the numerical modeling using ANNs created specifically in this project including the ANN model setup, calibration, analysis of the model performance and verification. The capability of the proposed numerical models to simulate wave height transformation over submerged breakwaters is investigated in Section 6. ANN models are tested in the numerical simulations and the results compared to the laboratory experimental data. And finally, Section 7 completes the paper with the main conclusions.

191

where xi, yi and yi are respectively input, target response and output computed by the network for the ith sample. A back-propagation (BP) algorithm (Rumelhart et al., 1986; Werbos, 1988) is usually employed to train MLP networks. The BP algorithm initializes the network parameters, propagates the input data forward through the network, and computes the output error as (Haykin, 1999): ⌢: ei ¼ yi −y i

ð3Þ

The computed error ei is then propagated backward through the network and the weights and biases are adjusted so as to minimize the mean-square error. Dealing with a nonlinear regression problem, batch mode is implemented to adjust the network parameters in BP learning on an epoch by epoch basis where each epoch consists of the entire set of training samples (Haykin, 1999). Although simplicity and efficiency in computations are significant advantages of BP learning, limitations on convergence and slow learning speed, which is even more annoying when complexity increase, is a disadvantage of the algorithm. For more information about MLP networks and BP algorithm, please see: Haykin (1999).

2. Artificial neural networks 2.2. Radial-basis function (RBF) network Artificial neural networks imitate the structure of biological neural networks. They consist of a massive number of computational units simulating neurons (acquiring knowledge) and connected together with interconnection weights which simulate synapses (storing the knowledge) (Haykin, 1999). Through a learning process, network parameters such as interlayer weights and bias values are modified and adapted. Depending on how the ANNs are applied, the learning process employed can be classified as supervised or unsupervised. Basically, supervised learning is suitable for function estimations when input–output paired data and a mathematical method for the learning process are available, while unsupervised learning may be best in the case of pattern recognition using data without a target (Haykin, 1999). In this research, we are dealing with a function estimation problem; therefore supervised learning is considered appropriate and will be discussed further. For more information about other learning methods and unsupervised methods in particular, please see: Haykin (1999). In engineering applications, the most popular artificial neural network is the multi-layer perceptron (MLP). Radial basis function (RBF) networks with universal non-linear approximation properties are similar to multi-layer perceptron in that they also use memory-based learning algorithms for their design (Haykin, 1999; Park and Sandberg, 1993). An introduction to ANN models, their basic framework, theoretical properties, and differences between the ANN models employed in this paper, are set out below.

The RBF neural network concept was introduced by Broomhead and Lowe (1988). Comparable to a MLP network, for the RBF network the entire data set is split into two main data sets and exploited by two distinct procedures of training and testing. The training process in RBF can be interpreted as a curve-fitting problem, to find a surface in multidimensional space which provides a best fit to the training data set. The testing process, known also as generalization, is to use this multidimensional surface for interpolating the test data set (Broomhead and Lowe, 1988; Poggio and Girosi, 1990). A RBF network consists of an input layer utilized to feed the input data into the network, a hidden layer to compute the outcome of the radial-basis functions and an output layer to combine the outputs from the radial-basis linearly. The outcomes of the input layer are calculated based on the Euclidean distance between the network input vector and RBF unit's center vector c. The hidden layer is a linear combination of basis functions and outputs of this layer that are weighted forms of the input layer outputs. A general expression of the network can be presented as (Robert and Howlett, 2001): yðxÞ ¼

Xm k¼1

wk Φðkx−ck kÞ þ b

where w1,w2,…,wm are weight factors, Φ is the hidden layer transfer function, ck is the center vector of the kth RBF unit and b is bias value. The most popular and widely used radial basis function is Gaussian:

2.1. Multi-layer perceptron (MLP) network



A multi-layer perceptron has an input layer, one or more hidden layers and an output layer of computation units named neurons. The transfer function of a multi-layer perceptron computes the inner product between the input vector and the corresponding inter-layer weight vector. Using a supervised form of learning for a given training data set S including n input–output pairs, the learning algorithm computes weights and biases of the network, so that the mean-square error (MSE) is minimized (Haykin, 1999): S ¼ fðxi ; yi Þg; i ¼ 1; …; n

ð1Þ

n  X ⌢ 2 yi −y i

MSE ¼

i¼1

n

ð2Þ

ð4Þ

−

kx−ck k2

Φðkx−ck kÞ ¼ e

2σ 2 k

 ð5Þ

where σk is the width of the kth RBF unit, a positive real number known also as the scaling parameter. 3. Data collection This section describes preparation of the experimental data set used in this study to train and test the ANN models. Preparing data in ANN modeling is an essential stage. For an accurate prediction, data preparation should be undertaken firstly to speed up the training process, secondly to reduce the model complexity, and thirdly to increase its scope for generalization. Therefore, achieving success in ANN modeling depends critically on this stage and requires that the ANN is trained using high quality data. In the present study, a large number of 3D experiments have been conducted in wave tanks in the presence of

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Table 1 The range of dimensional and non-dimensional data: training and interpolation testing data sets (Set 1, 2), extrapolation testing data set (Set 3) and larger scale data set (Set 4). Parameter

Mean

Set no.

1, 2

3

4

1, 2

3

4

1, 2

3

4

1, 2

3

4

Dimensional data Hi (mm) Lo (mm) hs (mm) h (mm) B (mm) G (mm) x (mm) y (mm)

46 1514 31 222 195 740 19 566

57 2454 76 265 195 740 33 563

167 2206 250 1000 250 2500 −375 1365

21 485 24 25 0 0 256 452

27 991 44 43 0 0 255 489

80 1667 0 0 0 0 707 1771

12 976 0 188 195 740 −280 −280

10 976 0 188 195 740 −280 −280

36 694 250 1000 250 2500 −1375 −1625

90 2126 60 250 195 740 490 1280

113 3380 112 300 195 740 530 1740

298 6243 250 1000 250 2500 625 4125

7.09 0.23 2.01 4.70 0.38 0.02 0.29

3.88 0.17 2.16 6.49 1.78 −0.27 0.97

2.13 0.10 0.96 3.36 0.17 0.20 0.37

2.51 0.14 2.03 3.27 0.20 0.15 0.32

2.00 0.08 1.64 4.92 1.03 0.60 1.56

3.85 0.05 0.00 2.16 0.35 −0.29 −0.29

3.55 0.03 0.00 1.72 0.22 −0.26 −0.29

1.53 0.04 0.84 2.52 0.40 −1.98 −2.34

13.36 0.47 5.04 16.36 0.76 0.50 1.31

14.68 0.60 11.25 19.50 0.76 0.54 1.78

13.17 0.30 6.94 20.83 3.60 0.90 5.94

Non-dimensional data 6.24 ξo Hi/h 0.21 0.87 hs/Hi 5.54 B/Hi 0.54 G/Lo 0.01 x/Lo 0.42 y/Lo

Standard deviation

Min

submerged breakwater structures over wide ranges of wave height, period, water depth, submergence depth and distance from the beach. Laboratory experimental data were collected in two wave flumes. This gave a range of different scales for each experiment. To properly investigate the contribution of 3D influences (diffraction) on the wave transmission behind the breakwater, impermeable breakwater models were constructed to prevent filtration through the structure. Therefore, this restricted wave transmission to overtopping and diffraction. Laboratory data sets collected are listed below: Data sets: (a) Set 1 (Training data): This data set is from a wave flume of 20 m long, 1.2 m wide and 1 m deep with a 1:15 sloping beach at one end. Experiments included various submergence depths, water depths and wave conditions. The data collected in this set were used for ANN model training. (b)Set 2 (Interpolation testing data): This data set was collected in the same flume as Set 1, and within the same range of physical parameters including submergence depths, water depths and wave conditions. The data collected in this set were used to test the ANN model in the case of interpolation prediction. (c) Set 3 (Extrapolation testing data): This data set includes the data collected in the same flume as Set 1, but with a different range of physical parameters including submergence depths, water depths and wave conditions. The data collected in this set were used to test the ANN model in the case of extrapolation prediction.

Max

(d) Set 4 (Larger scale testing data): This set of data includes the data collected in a 20 m long, 2.5 m wide and 1.2 m deep large tank with parabolic beach at the end. These data were used to validate the ANN model at a larger scale than the previous data sets, using different laboratory facilities and over a more diverse range of input parameters. Table 1 shows the range of dimensional and non-dimensional data including training and interpolation testing data sets (Sets 1, 2), extrapolation testing data set (Set 3) and larger scale data set (Set 4). Waves generated in both flumes were regular waves or monochromatic waves. Fig. 1 shows a plan and section of the smaller wave flume with the breakwater over the sloping beach at the end of the flume. Fig. 2 shows a plan and section of the large wave flume with the breakwater. Figs. 1 and 2 also indicate the positions of wave probes along and across the flumes. The next step is to determine the variables for the model. An ideal model is one which explains the problem in the easiest and simplest way using the fewest variables. It is important to identify the variables that save modeling time and space (Stein, 1993). As a first attempt eleven input parameters were considered: Incident wave height Hi, offshore wave length Lo, water depth over the breakwater crest hs, water depth in the toe of the breakwater h, distance from the shoreline d, breakwater crest width B, the seaward slope of the breakwater α, breakwater crest length l1, gap length l2 and Cartesian coordinates (or rectangular coordinates) of x and y. Preliminary architecture of the neural network was based on

Fig. 1. Wave flume layout and wave probe positions for a 3D test with 19 cm breakwater crest width, 30 cm water depth at the toe of the breakwater and 11.2 cm submergence depth (all measurements in the figure in cm).

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the initial data selection and the final number of input parameters selected on the basis of sensitivity to each parameter. Experimental data analysis done on the dimensional and non-dimensional parameters showed that the most important physical parameters could be restricted to fewer variables as listed in Tables 1 and 2 for dimensional and non-dimensional parameters respectively. The tables present the range of data used in the neural network models including minimum, maximum, mean and standard deviation of each individual parameter. For further analysis, on the basis of the dimensional analysis, it is assumed that the 3D wave transmission coefficient Kt can be defined as a function of the following dimensionless parameters:   H h B x y : K t;3D ¼ f ξo ; i ; s ; ; ; h H i H i Lo Lo

ð6Þ

In order to represent the combined effect of the wave steepness (so = Hi / Lo) and the seaward slope of the breakwater α, the surf similarity parameter or Iribarren number ξo is introduced as below: tanðαÞ ξo ¼ sffiffiffiffiffi : Hi Lo

ð7Þ

Therefore these nondimensional parameters can be classified as below: (a) The hydraulic parameters: the surf similarity parameter ξo and the relative wave height Hi/h, (b) The structural parameters: the submergence and crest width ratio hs/Hi and B/Hi, (c) The spatial parameters: the dimensionless Cartesian coordinates x/Lo and y/Lo. Applying dimensionless parameters helps to solve problems concerning the scale effects. However, to obviate the scaling problem entirely throughout the training, validation and future application of the ANN models, a normalization process is also essential. This can be achieved by:

xn ¼

x− minðxÞ maxðxÞ− minðxÞ

where x represents the parameter to be normalized.

ð8Þ

193

4. Classical approaches Developing a 3D design tool for prediction of global wave transmission coefficient Kd,t behind low-crested structures, Vicinanza et al. (2009) proposed a method based on a combined diffractionovertopping scheme for the first time. They proposed to employ the available wave diffraction theories combined with a 2D wave transmission formula. They assumed that wave diffraction occurred in the same way as for non-overtopped structures with no statistical correlation with 2D wave overtopping. Vicinanza et al. (2009) mentioned that the wave diffraction and 2D wave overtopping are totally different physical processes therefore, the hypothesis that they are uncorrelated seems quite reasonable. This gives, by summing energies: K d;t ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 2d þ K 2t

ð9Þ

where Kd and Kt are respectively wave diffraction coefficient and the two dimensional wave transmission coefficient. Vicinanza et al. (2009) recommended the solution by Penny and Price (1952) for the calculation of wave diffraction coefficients in comparison with Goda et al. (1978) method in particular. They mentioned that the method provided by Goda et al. (1978) is expressly valid for multidirectional seas or random (short-crested) waves; however, the method of Penny and Price (1952) relies on linear periodic wave theory and is used for monochromatic waves. In addition, although the model of Penny and Price (1952) is complicated by the presence of Fresnel integrals, it is much more easy to use in comparison with Goda et al. (1978) particularly for the initial stages of design (Vicinanza et al., 2009). Kd is calculated by Penney and Price (1952) based on the penetration of waves through a gap smaller than one wave length. When gap width is small compared with wave length, an approximate solution may be obtained by assuming that the streamlines for the motion of the water through the gap are the same as for a simple uniform streaming of water through the gap (Penney and Price, 1952). Penney and Price (1952) mentioned that the mathematical problem then becomes identical with the problem of the penetration of sound waves through a narrow slit; this problem is treated by Lamb (1924). Using Lamb's solution, the following expression for F(x, y) for points not too near the gap can be deduced (Penney and Price, 1952):   1 ðLo =4rÞ exp − πi−kri 4 F¼ 1 logðπb=4Lo Þ þ γ þ πi 2

ð10Þ

where r is the distance from the center of the gap, b is the gap width and γ is Euler's constant, equal to 0.5772.

Fig. 2. Plan of large wave tank with wave probe positions for a 3D test with 75 cm breakwater crest width, 100 cm water depth and 25 cm submergence depth (all measurements in the figure in cm).

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Table 2 Comparison of the available 2D empirical formulae (Buccino and Calabrese, 2007; d'Angremond et al., 1996; Goda and Ahrens, 2008 and van der Meer et al., 2005) and 3D approach (Vicinanza et al., 2009) based on models by Goda and Ahrens (2008) (transmission) and Penney and Price (1952) (diffraction) for wave transmission coefficient behind submerged breakwaters with experimental data for different measurement points (for: Hi = 55 mm, Lo = (a): 976, (b): 1427 and (c): 3380 mm, hs = 0 mm B = 195 mm and h = 188 mm). x (mm)

y (mm)

x/Lo

y/Lo

Kt,2D-D

Kt,2D-VDM

Kt,2D-BC

Kt,2D-GA

Kd-PP

Kt,3D-V

Kt,3D-M

(a) 449 481 282 476 303 108 9 0 −10 −107 −256 −249 −250 −250

282 819 528 1302 1049 820 280 530 1050 820 1051 530 280 31

0.46 0.49 0.29 0.49 0.31 0.11 0.01 0.00 −0.01 −0.11 −0.26 −0.26 −0.26 −0.26

0.29 0.84 0.54 1.33 1.07 0.84 0.29 0.54 1.08 0.84 1.08 0.54 0.29 0.03

0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47

0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.44

0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28

0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34

0.29 0.21 0.27 0.18 0.20 0.23 0.40 0.29 0.20 0.23 0.20 0.27 0.34 0.42

0.40 0.35 0.39 0.33 0.34 0.36 0.49 0.40 0.20 0.23 0.20 0.27 0.34 0.42

0.24 0.27 0.19 0.13 0.28 0.19 0.30 0.22 0.14 0.18 0.36 0.31 0.43 0.57

(b) 449 481 282 476 303 108 9 0 −10 −107 −256 −249 −250 −250

282 819 528 1302 1049 820 280 530 1050 820 1051 530 280 31

0.31 0.34 0.20 0.33 0.21 0.08 0.01 0.00 −0.01 −0.07 −0.18 −0.17 −0.18 −0.18

0.20 0.57 0.37 0.91 0.74 0.57 0.20 0.37 0.74 0.57 0.74 0.37 0.20 0.02

0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50

0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47

0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32

0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38

0.37 0.28 0.35 0.23 0.26 0.30 0.51 0.37 0.26 0.30 0.26 0.35 0.44 0.54

0.49 0.43 0.47 0.39 0.41 0.44 0.60 0.49 0.26 0.30 0.26 0.35 0.44 0.54

0.20 0.29 0.24 0.14 0.29 0.30 0.44 0.35 0.33 0.46 0.50 0.54 0.47 0.66

(c) 449 481 282 476 303 108 9 0 −10 −107 −256 −249 −250 −250

282 819 528 1302 1049 820 280 530 1050 820 1051 530 280 31

0.13 0.14 0.08 0.14 0.09 0.03 0.00 0.00 0.00 −0.03 −0.08 −0.07 −0.07 −0.07

0.08 0.24 0.16 0.39 0.31 0.24 0.08 0.16 0.31 0.24 0.31 0.16 0.08 0.01

0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53

0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50

0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39

0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48

0.65 0.49 0.61 0.40 0.45 0.52 0.89 0.65 0.46 0.52 0.45 0.62 0.77 0.94

0.76 0.63 0.72 0.56 0.60 0.65 0.97 0.76 0.46 0.52 0.45 0.62 0.77 0.94

0.28 0.26 0.20 0.14 0.29 0.36 0.38 0.42 0.45 0.52 0.55 0.60 0.70 0.90

This corresponds to waves diverging uniformly in all directions behind the breakwater from the gap as a point source, the ratio of diffracted wave height at any point to incident wave height being given by mod F, which is proportional to r−1/2 (Penney and Price, 1952). The expression for the Kd (mod F) may be written in the form (Penney and Price, 1952): π K d ¼ modF ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi

1 2 2 kb log 18 kb þ γ 2 þ π 4

sffiffiffiffiffiffiffiffiffiffiffiffi  ffi b πr

ð11Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi and the coefficient of ðb=πr Þ in this expression is slightly greater than unity when b lies between 0.1Lo and 0.2Lo. Thus when b = Lo / 2π the value of the coefficient is 1.053. Hence for this range of values of b, Kd pffiffiffiffiffiffiffiffiffiffiffiffiffiffi is given approximately by ðb=πr Þ (Penney and Price, 1952). In order to investigate the capability of existing empirical formulae and to highlight the importance of a 3D design tool, analysis is presented below using methods selected from the literature.

The 2D empirical formulae presented by d'Angremond et al. (1996), van der Meer et al. (2005), Buccino and Calabrese (2007) and Goda and Ahrens (2008) are applied and the wave transmission coefficients computed using these formulations will hereinafter be referred to as Kt,2D-D, Kt,2D-VDM, Kt,2D-BC and Kt,2D-GA respectively. The 3D method of Vicinanza et al. (2009) is also applied for comparison (Kt,3D-V). The two dimensional wave transmission coefficient (Kt,2D) is calculated using the method of Buccino and Calabrese (2007) and the wave diffraction coefficient (Kd) is calculated using the method proposed by Penney and Price (1952). Then, using the computed values Kt,2D and Kd, the three dimensional wave transmission coefficient (Kt,3D) is calculated by combining the two. In order to perform a meaningful comparison, the selected approaches are tested using experimental data within the same range the models were calibrated. Table 2 presents a comparison between the present measurements and 2D and 3D predictions for some specific wave conditions (Hi = 55 mm and Lo = 976, 1427 and 3380 mm) and hs = 0 mm, B = 195 mm and h = 188 mm at different measurement points behind 3D submerged breakwaters. The variation of wave transmission

A. Sharifahmadian, R.R. Simons / Coastal Engineering 83 (2014) 190–204

coefficient behind the structure is very significant; this contrasts with predictions from the 2D models which provide just a single value for Kt without full information about its spatial variation. This is the main weakness of these 2D models. A comparison has been made between the predictions of transmission coefficient (Kt,3D-V) obtained from the model by Vicinanza et al. (2009) and the corresponding observed data (Kt,3D-M) (Table 2). The comparison indicates that the model does not agree well with the experiments. Although the disagreement between the results using the method of Buccino and Calabrese (2007) and measurements leads to an error in the calculation of (Kt,2D) and consequently of (Kt,3D) using the method of Vicinanza et al. (2009), the more significant error in determining Kd using Penney and Price (1952) is the main source of error in calculating Kt,3D using the Vicinanza et al. (2009) method. The Kd computed using the Penney and Price (1952) method (Kd-PP) is not able to provide full information about the variation of Kd behind the structure. The method does not include the effects of different physical parameters such as incident wave height, water depth and breakwater geometry. Kd is calculated based only on the offshore wave length and the radial distance from the center of the gap using a very simplified method. Another significant source of error in determining the three dimensional wave transmission coefficient using the Vicinanza et al. (2009) method is the way it uses to combine the wave diffraction theories with 2D wave transmission formulae. As mentioned before, Vicinanza et al. (2009) assumed that wave diffraction occurred in the same way as for non-overtopped structures with no statistical correlation with 2D wave overtopping. The hypothesis that wave diffraction and 2D transmission are uncorrelated seems quite reasonable for lowcrested structures, as they are totally different physical processes (Vicinanza et al., 2009); however, it seems that it is not very accurate in the case of submerged breakwaters as the wave diffraction and 2D transmission seem to be correlated for fully submerged breakwaters. Therefore, using the method of Vicinanza et al. (2009), the results obtained are significantly over-estimated. It should be mentioned that the Vicinanza et al. (2009) method was originally proposed for lowcrested breakwaters and not for fully submerged breakwaters. In addition, the hypothesis that wave diffraction and transmission are un‐correlated may hold for random waves (for which the method is strictly valid), but not for regular waves. These analyses confirm the significance of 3D effects around submerged breakwaters and the need for an accurate and reliable 3D design tool. 5. ANN model setup and performance A brief introduction and explanation of prediction models based on ANNs was presented in Section 2. This section concerns the numerical modeling using ANNs created specifically in this project to predict the wave transmission behind the submerged breakwater. 5.1. Model calibration While the number of layers in MLP can be variable, RBF networks are always designed with three layers of neurons. The input and output layers in RBF are the same as in MLP, while the hidden layers are not. The main difference is related to the transfer functions applied in the hidden layer in these two models. The RBF transfer functions compute the distance of the input vector to the center of that hidden unit known as the Euclidean norm, while a MLP unit applies a transfer function which computes the inner product of the weight and the input vectors. Both RBF and MLP models include biases in their hidden and output layers. However, RBF networks usually have more computational units in the hidden layer than MLP as their transfer functions respond to a smaller area of the input space than MLP. This is of benefit as it requires less training time to design RBF compared with MLP. In the present

195

study the neurons in the different layers were arranged to learn at the same rate by assigning a learning rate parameter. The selection of the optimum network is based on a trial and error scheme. A predefined rule does not exist for the selection of the number of hidden layers. The performance of the ANN scheme depends critically on the quality and the quantity of the data. Insufficient data and existence of noise in the data can lead to a reduction in the efficiency of the network in predicting the unknown parameters of the phenomenon under study. A cross-validation method (Hagan et al., 1996) was employed in this study to deal with the problem of over-fitting during the training process and to improve the generalization for future applications of the model. This involved dividing the input data into two subsets, one for model training and the other for final testing. The set of training examples was split into two parts: estimation and validation subsets. The estimation subset contains data used to train the ANN models, while the validation subset was used to evaluate the model performance and monitor the training process. The training samples were set up to be completely random for an improved learning procedure. The validation error normally decreases during the initial phase of training, as does the training set error. However, when the network begins to overfit the data, the error in the validation set typically begins to rise. The network weights and biases were saved at values giving the minimum error from the validation set. Having been tuned using the entire training data set the network was verified and the different models compared using the test data not seen before. The computed error in the test data was scrutinized during the training process. If the error in the testing data reaches its lowest value at an iteration number with a significant difference from the validation data, this might imply a substandard data division. The philosophy behind employing test data is to test the trained neural network using data not used in the validation or training stages. This examination of the performance and capability of the model under new conditions ensures that it works properly and is not subject to generalization problems. The data set described in Table 2 was used to train the networks and to investigate the model performance. The learning algorithm and modeling process used for the networks are described below. Although a higher number of hidden layers might increase the generalization performance of MLP, it increases the computation time when a back propagation algorithm is being used for training. Therefore, in this study, it is assumed that one hidden layer is sufficient to approximate the underlying function adequately. For a more detailed and formal description of MLP neural networks, the reader is referred to Hagan et al. (1996) or Haykin (1999). Once the number of hidden layers in the network is fixed, there remain a few parameters to be determined, usually experimentally. This also means that often a large parameter space has to be searched. For standard MLP networks, the size of the hidden layer, the learning rate, the activation function and the minimum gradient are the most important parameters that have to be set. An antisymmetric hyperbolic tangent transfer function was used in the MLP as it gave a better performance in BP learning in comparison with other functions such as asymmetrical logistic transfer functions. In RBF model training, initially the hidden layer has zero number of neurons. To simulate the network for all training samples, the computed output is compared to the target value of the respective sample. A RBF unit is added to the hidden layer with weights equal to the input vector with the maximum error. The connection weight parameters between the hidden and the output layers are adjusted to minimize the error. This error which is statistically described as the mean squared error must be lower than the error goal parameter. Other parameters have to be met before training is complete. The radius parameter should be selected large enough to allow the hidden neurons to respond properly to overlapping regions of the input space. However, choosing a very large value might cause all the neurons to respond in a similar manner. At each iteration the input vector that gives the lowest network error is used to create a radial basis neuron. The error produced by the updated network is checked, and if it is lower than the error goal training is stopped; otherwise, one more neuron is added to the hidden layer.

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This procedure is repeated until the error goal is met or the maximum number of neurons is reached. The training of the RBF model was terminated once the calculated error reached the goal of 1% or the maximum number of training iterations (500) had been completed. In the case of the MLP network, training was terminated once the number of iterations reached 3000 or when the maximum output error was reduced to below 1% of the target value. In all cases the weights that corresponded to the minimum error during the training stage were taken as the final model parameters. In the training stage for the MLP model, hidden layers with 9 nodes gave the least errors; hence 9 hidden units were chosen to represent the MLP model's hidden layer. In the case of the RBF model, a 25-noded hidden layer network was chosen. 5.2. Model verification The predictions of transmission coefficients obtained by MLP and RBF models plotted against the corresponding observed data (Fig. 3) show that they are comparable, with good agreement and some limited exceptions. The RBF model performs particularly well. The comparison of the two ANN models with the experimental data was investigated

Table 3 Statistical comparison between predictions from RBF and MLP models by multiple correlation coefficient and root mean squared error. Statistical index

MLP

RBF

R2 RMSE

0.96 0.09

0.98 0.06

by comparing the two statistical parameters, root mean squared error (RMSE) and the squared multiple correlation coefficient R2 (Table 3): ! 31 2 XN X c −X m 2 2 i i 7 6 i¼1 X mi 7 6 7 RMSE ¼ 6 7 6 N 5 4

ð12Þ

X   2 N X mi −X m X ci −X c i¼1 R ¼X  2 X  2 : N N X mi −X m X ci −X c i¼1 i¼1 2

ð13Þ

1

(a) +10% 0.8

Kt (MLP)

−10% 0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

Kt (Measured)

Both graphical and statistical comparisons indicate a better performance from the RBF model than for the MLP model. Much higher squared multiple correlation coefficient and lower root mean squared error confirmed the better performance of the RBF model. The error distribution is also shown in Fig. 4, using notched box-andwhisker plots (McGill et al., 1978; Tukey, 1977) indicating the degree of dispersion (spread) and skewness in Kt, as well as identifying outliers. Basically, the box plots display differences between groups of non-parametrically data without making any assumptions as to the underlying statistical distribution using five distinct numbers. The bottom and top of the box are the 25th and 75th percentiles (the lower and upper quartiles respectively). The distance between the top and bottom of the box is the interquartile range and the band near the middle of the box is the 50th percentile (the median). If the median is not centered in the box, it shows sample skewness. The whiskers are lines extending above and below each box. Whiskers are drawn from the ends of the interquartile ranges to the furthest observations within the whisker length (the adjacent values). Any data not included between the whiskers are plotted as an outlier with a “+” marker. An outlier is a value that is more than 1.5

50

1

(b)

40

+10% 30

0.8

Percentage error (%)

Kt (RBF)

−10% 0.6

0.4

0.2

20 10 0 −10 −20 −30 −40

0

0

0.2

0.4

0.6

0.8

1

Kt (Measured) Fig. 3. Graphical comparison between predictions from MLP and RBF models by the scatter plot of the estimated and the observed values of Kt.

−50

MLP

RBF

Fig. 4. Descriptive comparison between predictions from MLP and RBF models, by the distribution of error between the estimated and the observed values of Kt using notched boxand-whisker plots.

A. Sharifahmadian, R.R. Simons / Coastal Engineering 83 (2014) 190–204

1

197

Table 4 Statistical comparison of non-dimensional and dimensional RBF models.

+10% 0.8

Statistical index

Dimensional model

Non-dimensional model

R2 RMSE

0.98 0.10

0.98 0.06

Kt (Calculated)

−10% 0.6 validated against the experimental data for three different applications and its performance assessed in following section.

0.4

6. Validity and reliability of the prediction model

0.2

Dimensional RBF network Non−dimensional RBF network 0 0

0.2

0.4

0.6

0.8

1

Kt (Measured) Fig. 5. RBF model results: dimensional and non-dimensional networks.

times the interquartile range away from the top or bottom of the box (Tukey, 1977). Notches display the variability of the median between samples. The width of a notch is computed so that box plots whose notches do not overlap have different medians at the 5% significance level. The significance level is based on a normal distribution assumption (McGill et al., 1978). Error notched box-and-whisker plots in Fig. 4 illustrate the better performance of the RBF model considering each error data category. Similar graphical and descriptive comparisons were made between the non-dimensional and dimensional models (Figs. 5 and 6) as well as statistically in Table 4. Although both models perform well, slightly better performance can be seen in the non-dimensional model. This can also be observed from the statistical indexes. Based on all comparisons made in this section, a non-dimensional radial basis functional (RBF) neural network was selected as the best model to simulate wave transmission coefficients around submerged breakwaters. This is

50

To evaluate the performance of the trained ANN model, it was applied over three ranges. These included interpolation, extrapolation, and prediction at larger scale well beyond that used in training the model. To demonstrate the influence of the input variables on the ANN's predictions a sensitivity analysis was conducted. The next section considers these analyses. Interpolation predictions were done within the range of input parameters used for training the model as defined in Table 2 (Set 1). The neural networks are basically trained within a specific range of inputs and are expected to return acceptable results for new values of input parameters within the same range (Set 2). However, a trained model is likely to be employed to predict wave transmission for input values which are not necessary within the range of the training data set. Detailed analysis of extrapolation predictions was conducted for values of input parameters outside the range of the training data set (Set 3). This refers both to the wave conditions and to the spatial location of the observation. A separate set of conditions was identified to investigate predictions at a far larger scale as it is important to have an understanding about the performance of the model when the scales increase significantly. This might be critical if the neural network model is being trained with laboratory data and is going to be used to predict real sea states. The data used in this case were those in Set 4. Finally, a sensitivity analysis was conducted to study the effect of each input parameter on model performance. 6.1. Accuracy analysis Comparison of the proposed model with the experimental data was investigated by comparing five statistical parameters including the root mean squared error (RMSE) and the squared multiple correlation coefficient R2 as defined in previous section, and three additional parameters: bias or distortion β, Wilmott index Iw (Wilmott, 1981) and error function ε (Haller et al., 2002):

40

N  2 X X ci −X mi

Percentage error (%)

30

Iw ¼ 1−

20

i¼1



2 N  X



X ci −X m þ X mi −X m

ð14Þ

i¼1

10

N X X ci

0 β¼

−10

X mi

−30 −40

ð15Þ

N

31 = 2 2 2 N  7 6X 7 6 X ci −X mi 7 6 7 6 ε ¼ 6 i¼1 7 7 6 7 6 N X 4 2 5 X mi

−20

−50

i¼1

!

ð16Þ

i¼1

Dimensional RBF

Non−dimensional RBF

Fig. 6. RBF model results: dimensional and non-dimensional networks.

where N represents the number of data in each group, Xc is the calculated value, and Xm is the measured value and the barred parameters

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13

1

9

8 +10%

11

8

12.6

−10%

8

5

0.6

10

9

7

0.8

6 7 8

8

9

Y(m)

10

0.4

12.2

7

9

Kt (Calculated)

10

10

11.8

0.2

0.4

0.6

0.8

1

11.4

Fig. 7. Correlation of RBF and experimental values for interpolation cases (data set: Set 2; test with: Hi = 22 mm, 41 mm, 60 mm, 81 mm, T = 1 sec, hs = 62 mm, h = 250 mm).

12 0.4

0.8

8

0

11

8

Kt (Measured)

9 10

0.2

7

0 0

9

10

1.2

X(m) Fig. 8. Error contour map of RBF for interpolation cases (data set: Set 2; test with: Hi = 22 mm, 41 mm, 60 mm, 81 mm, T = 1 mm, hs = 62 mm, h = 250 mm).

demonstrate the average values of the parameters. Perfect agreement is achieved if R2 and Wilmott index are 1.0, the error function is zero, the value of distortion approaches 1.0 and RMSE is small. R2 is considered unsatisfactory as a measure of the goodness of fit for a multivariate regression relationship (Draper, 1984) and thus the validity of the model will be assessed mostly by the other parameters. The values of the five statistical parameters are tabulated and compared graphically below. The proposed ANN model was validated with new experimental data not used during the training process. Accuracy analysis was performed for the three different scales of application referred above and explained below. 6.1.1. Interpolation A statistical comparison was made of wave transmission coefficients around the submerged breakwater for cases involving interpolation. Conditions are as set out in Table 1. Fig. 7 shows the correlation between predicted and measured values within different regions around the breakwater for a case with input parameters within the range of the training data set. A specific wave period of 1 s and total water depth and submergence depth of 250 mm and 62 mm respectively were employed in this test. Incident wave height varied from 22 mm up to 81 mm. The values of the statistical indicators and the graphical comparison both indicate that the ANN model is able to calculate wave transmission coefficient with a high accuracy. The predicted values gave high correlation (R2 = 97%) and Wilmot number (Iw = 74%) and low values of ε (0.23) and RMSE (0.04) as well as a value of distortion (β) close to 1.0, indicating very good accuracy (Table 5). Fig. 8 shows an error contour map for the data plotted in Fig. 7; this indicates the absolute percentage error in different regions around the submerged breakwater. As can been seen, the non-dimensional RBF model gives low error (mostly less than 10%) in most regions behind

Table 5 Statistical evaluation of non-dimensional RBF model (data set: Set 2). Statistical index

Iw

ε

R2

β

RMSE

Non-dimensional RBF model

0.74

0.23

0.97

0.99

0.04

the breakwater. Higher values of error up to 12% can be seen in some small regions (i.e. far end of the shadow region of the structure and over the breakwater roundhead). Additional analysis was also carried out to examine the ability of the proposed model to predict Kt in various regions behind the breakwater as spatial prediction of Kt is the primary interest of this study. Fig. 9 presents the results of this analysis for three different points behind the structure. Each graph illustrates the variation of Kt for different submergence ratio at one specific point. The general trend of Kt computed by the ANN model in each graph follows the pattern of measured values reasonably. As expected, Kt increases with submergence ratio for points behind the breakwater (Fig. 9(a) and (c)). Kt decreases at points right behind the breakwater trunk (2D transmitted waves only; Fig. 9(a): x = 450 mm and y = 240 mm) compared with points located in the gap (diffracted waves only; Fig. 9(b): x = −30 mm and y = 500 mm) because of wave dissipation by breaking over the crest. However, an increase in Kt can be observed in the point at x = 470 mm and y = 760 mm (combined 2D transmitted and diffracted waves; Fig. 9(c)) compared with the point right behind the breakwater trunk (Fig. 9(a)) due to the effect of wave diffraction. It can be also deduced from the plots that the model produces reasonable responses under different conditions and locations behind the breakwater, particularly, at points located in the shadow region of the structure, with errors up to a highest value of 10%. 6.1.2. Extrapolation Data set 3 was employed to test the proposed model in cases involving extrapolation. The parametric range of data is presented in Table 1. Fig. 10 shows the correlation between predicted and measured values within different regions around the breakwater for a case with input parameters (wave period and water depth over and in front of the structure) within the range of the training data but incident wave heights outside the range of the training data. A specific wave period of 1 s and total water depth and submergence depth of 250 mm and 62 mm respectively are employed in this test. Incident wave height varied from 88 mm up to 100 mm. The values of the statistical indicators and graphical comparison indicate that the ANN model is able to predict

A. Sharifahmadian, R.R. Simons / Coastal Engineering 83 (2014) 190–204

1

1

(a)

(b)

0.8

0.8

Kt

Kt

199

0.6

0.4

0.6

0.4 x = −30 mm y = 500 mm

x = 450 mm y = 240 mm

0.2

0

1

2

3

0.2

4

0

1

2

3

4

hs/Hi

hs/Hi 1

(c)

Kt

0.8

0.6

0.4 x = 470 mm y = 760 mm

0.2 0

1

2

3

4

hs/Hi Fig. 9. Spatial wave transmission coefficient Kt versus submergence ratio hs/Hi at different locations: measured values (dots) and calculated values by proposed ANN model (solid line).

wave transmission coefficients with high accuracy. The values calculated by the ANN provide acceptable values of R2 and Iw (greater than 75%) and low values of ε (0.26) and RMSE (0.15) as well as a distortion value of 0.99 that indicate good accuracy (Table 6). Fig. 11 is an error contour map of the same wave transmission calculated by the RBF model as plotted in Fig. 10. This shows the error in different regions around the submerged breakwater. The graph shows good agreements in most regions with lower values of the absolute percentage error in regions behind breakwater, increasing closer to the region over the breakwater roundhead. As can been seen, the RBF model performs with low error (mostly less than 15%) in most regions behind the breakwater. Less accurate results (errors higher than 15%) were found in a small region at the far end of the shadow region of the structure and a narrow band at the far end of the breakwater with the maximum error of 18%. Higher values of error up to 30% can be seen in a small region over the breakwater roundhead.

1

+10% 0.8

Kt (Calculated)

−10% 0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

Kt (Measured) Fig. 10. Correlation of RBF model results and experiments in the case of extrapolation (data set: Set 3; test with: Hi = 88 mm, 96 mm, 100 mm, T = 1 s, hs = 62 mm, h = 250 mm).

Table 6 Statistical evaluation of nondimensional RBF models (extrapolation; Set 3). Statistical index

Iw

ε

R2

β

RMSE

RBF model

0.75

0.26

0.77

0.99

0.15

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13

2 6 18 1 5 1 9

Table 7 Statistical evaluation of the nondimensional RBF model in larger scale model with a scale factor of 4.

9

6 15 12

12

12

12.6

3 3

6

6

9

15 15 15

8 8 1

12

0

18

24

21 15

11.4

18

18

18

30 27

0.4

ε

R2

β

RMSE

Larger scale model

0.73

0.39

0.85

1.05

0.19

Additional analysis is also carried out in this section to examine the ability of the proposed model to predict Kt including extrapolation to various regions behind the breakwater. Fig. 12 presents the results of this analysis for three different points behind the structure. Each graph illustrates the variation of Kt with surf similarity parameter for one specific point. The general trend of Kt computed by the model follows the experimental pattern reasonably well. As expected, Kt increases with surf similarity parameter for points behind breakwater (Fig. 12(a) and (c)). At point right behind the breakwater trunk (2D transmitted waves only; Fig. 12(a): x = 450 mm and y = 240 mm) compared with point located in the gap (diffracted waves only; Fig. 12(b): x = −30 mm and y = 500 mm), Kt decreases for ξo b 9 because of wave dissipation by breaking over the crest while Kt increases for ξo N 9 due to wave shoaling over the crest. In a similar way to Fig. 9, a slightly increase in Kt can be observed in the point at x = 470 mm and y = 760 mm (combined 2D transmitted and diffracted waves; Fig. 12(c)) compared with

1

12

Y(m)

12

11.8

Iw

9

15 12.2

Statistical index

0.8

1.2

X(m) Fig. 11. Error contour map of RBF model results and experiments for cases of extrapolation (data set: Set 3; test with: Hi = 90 mm, 93 mm, 96 mm, 100 mm, T = 1 s, hs = 62 mm, h = 250 mm).

1

1

(a)

(b) 0.8

Kt

Kt

0.8

0.6

0.6

0.4

0.4 x = 450 mm y = 240 mm

0.2

3

6

9

x = −30 mm y = 500 mm

12

15

0.2

3

6

9

ξo

12

15

ξo 1

(c)

Kt

0.8

0.6

0.4 x = 470 mm y = 760 mm

0.2

3

6

9

ξo

12

15

Fig. 12. Spatial wave transmission coefficient Kt versus the surf similarity parameter ξo at different locations: measured values (dots) and calculated values by proposed ANN model (solid line).

A. Sharifahmadian, R.R. Simons / Coastal Engineering 83 (2014) 190–204

1

Table 8 Statistical comparison of ANN model and Vicinanza et al. (2009) method (data sets: Sets 2, 3 and 4).

+10% 0.8

Kt (Calculated)

−10%

Statistical index

Iw

ε

R2

β

RMSE

Non-dimensional RBF model Vicinanza et al. (2009)

0.77 0.64

0.38 0.51

0.93 0.13

1.04 1.57

0.19 1.02

0.6 even in the extrapolation case the predictions are still acceptable (e b 15%).

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

Kt (Measured) Fig. 13. Graphical comparison of the non-dimensional RBF model extrapolated to larger scale (data set: Set 4; test with: Hi = 88 mm, 135 mm, 180 mm, 210 mm, 240 mm, T = 2 s, hs = 250 mm, h = 1000 mm).

the point right behind the breakwater trunk (Fig. 12(a)) due to the effect of wave diffraction. It can also be deduced from the plots that the model produces reasonable responses under different conditions and locations behind the breakwater, particularly, at points located in shadow region of the structure with errors up to a highest value of 15%. These graphs include the extrapolation cases where the input data are not within the range of the training data. Referring to Tables 2 and 8, values of surf similarity parameter larger than 13.36 are considered as extrapolation cases. The graphs show disagreements between predicted and measured values for ξo N 13.36 as the error increases for the extrapolation cases. However,

1

0.8

Kt (Calculated)

201

0.6

0.4

Kt, ANN: Diff.+Trans.

0.2

Kt, ANN: Diff. Kt, V: Diff.+Trans.

6.1.3. Larger scale A set of experimental data from the large flume was employed to assess the performance of the ANN model under larger scale conditions and using different laboratory facilities for cases involving input parameters lying both inside and outside that used for training (Table 1). The evaluation data for these larger scale tests involved a scale factor of 4. The statistical comparison is presented in Table 7 and Fig. 13 shows the correlation between predicted and measured values for these tests. The values of the statistical indicators and the graphical comparison indicate that the ANN model is able to calculate wave transmission coefficients with good accuracy. The values predicted by the ANN model provide fairly good R2 (85%) and Wilmot number Iw (73%). A low value of ε (0.39) and a convincing value of RMSE (0.19) were also obtained. A value of distortion (β) close to 1.0 also points to the good accuracy of the predictions (Table 7). 6.1.4. Classical approach In order to show the capability of the proposed ANN model, a comparison with an existing design tool is carried out. The method of Vicinanza et al. (2009) is applied for comparison. The predictions of transmission coefficients obtained by ANN and the model by Vicinanza et al. (2009) plotted against the corresponding observed data (Fig. 14) show the more accurate results from the ANN model. Comparison of the two models with the experimental data was also investigated by comparing the five statistical parameters (Table 8). Both graphical and statistical comparisons indicate a better performance from the ANN model than from the Vicinanza et al. (2009) model. A comparison of the predicted transmission coefficients (Kt,V) obtained from the model by Vicinanza et al. (2009) and the ANN model (Kt,ANN) with the corresponding observed data (Kt,M) for two main zones has been made; one is the shadow zone (the region sheltered by the breakwater) and the other one is the region located behind the gap. The comparison indicates that the Vicinanza et al. (2009) model does not agree well with the experiments for those points located in the shadow zone where both diffraction and overtopping affect the transmitted wave height as error is introduced by both the transmission and diffraction formulae. However, for those coming from the region behind the gap where the transmission coefficient does not apply and the errors are due uniquely to the diffraction method, the results of Vicinanza et al. (2009) model are in relatively good agreement with the experiments. As mentioned before, the significant source of error in determining three dimensional wave transmission coefficients using the Vicinanza et al. (2009) method is the way it combines the wave diffraction theories with 2D wave transmission formulae, assuming no statistical

Kt, V: Diff.

0 0

0.2

0.4

0.6

0.8

1

Kt (Measured) Fig. 14. Scatter plot of calculated and measured Kt for ANN and Vicinanza et al. (2009) models for two cases: considering only the diffraction effect (region behind the gap; shown with markers with solid fill) and combined diffraction and transmission effects (region sheltered by the breakwater; shown with markers without fill).

Table 9 Sensitivity analysis of RBF model (data set: Set 2). Omitted parameter

ξo

Hi/h

hs/Hi

B/Hi

x/Lo

y/Lo

R2 β RMSE

0.93 1.06 0.21

0.94 0.98 0.28

0.94 1.00 0.13

0.97 0.99 0.10

0.77 1.10 0.40

0.75 1.06 0.31

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correlation between wave diffraction and 2D wave overtopping. Basically, this hypothesis is quite reasonable for low-crested structures, as they are totally different physical processes (Vicinanza et al., 2009); however, it seems that it is not very accurate in the case of submerged breakwaters as wave diffraction and 2D transmission seem to be

correlated for fully submerged breakwaters. Therefore, using the method of Vicinanza et al. (2009), the results obtained are significantly overestimated in the shadow zone. It should be mentioned that the Vicinanza et al. (2009) method was originally proposed for surface‐piercing breakwaters and that the

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hypothesis that wave diffraction and transmission are un‐correlated may hold for random waves (for which the method is strictly valid), but not for regular waves. 6.2. Sensitivity analysis All parameters relevant to the wave field around a 3D submerged breakwater have been included in the ANN models described above. In order to detect the importance of each parameter to the performance of the model, networks were designed and trained using the same training data sets but with each parameter omitted in turn, in order that the most influential parameters could be identified. Assessments were made on the performance of the networks by comparison with the network results using three statistical parameters. This investigation illustrated that the dimensionless Cartesian coordinates x/Lo and y/Lo were the most significant parameter in the 3D wave field around the breakwater, with wave height and energy varying spatially around the structure. It can be seen that the spatial parameters x/Lo and y/Lo have the lowest R2 correlation coefficients (0.77 and 0.75 respectively) and the highest error (RMSE = 0.40 and 0.31) with bias (β) values of 1.10 and 1.06. This confirms the importance of 3D effects on wave height prediction and highlights the inadequacy of 2D models that are unable to deal with spatial variation of wave height behind the breakwater. From Table 9, it can be seen that the hydraulic parameters ξo and Hi/h have the next lowest R2 correlation coefficients (0.93 and 0.94 respectively) and the highest error (RMSE = 0.21 and 0.28) with bias (β) values of 1.06 and 0.98. These parameters also have a considerable influence on the results of the networks. It should be noted that the division into training data, validation data and test data for the sensitivity analysis was carried out in the same way as for the main ANN model. The R2 correlation coefficient, bias β and RMSE values for the main ANN model are 0.97, 0.99 and 0.04 respectively. A detailed sensitivity analysis is also presented in Fig. 15 comprising different sensitivity graphs of individual non-dimensional input parameters. The sensitivity figures contain several lines. The dash lines show the band of the 95% confidence interval, one line for the lower boundary (quartile = 2.5%) and the other for the upper boundary (quartile = 97.5%). The solid line is the ANN prediction (mean value). Fig. 15(a) reveals that the ANN model gives a relatively high reliability in the range of 3.75 b ξo b 9.15. Outside this interval the model tends to low reliability. It can be seen in Fig. 15(b) that for Hi/h the confidence band is narrow with high reliability between values 0.09 and 0.44 and with acceptable reliability outside this range for Hi/h values up to 0.53. Based on Fig. 15(c), the model is reliable in the range hs/Hi b 3.8. For larger submergence depth the ANN model is rather unreliable. According to this plot, Kt increases dramatically with submergence ratio from a value of Kt = 0.33 at zero submergence to a value of Kt = 0.67 at hs/Hi = 3.2. Beyond this point, no significant influence of the submergence ratio is seen. Fig. 15(d) shows that the model is also reliable in the range 1.9 b B/Hi b 7.3. There is a narrow confidence band with high reliability in Fig. 15(e) for − 0.22 b x/Lo b 0.42, and with acceptable reliability outside this range which suggests that the model is reliable across the entire range of − 0.3 and 0.5. The model only seems to be reliable for − 0.02 b y/Lo b 1.15; further from the breakwater, the model has a rather low reliability (Fig. 15(f)). 7. Conclusions Comparisons reveal the encouraging performance and reliable predictions from the ANN model. The following conclusions are offered: the performance of the ANN model with non-dimensional parameters was better than with dimensional parameters in prediction of wave transmission coefficient around the submerged breakwater. The ANN model based on RBF presents more acceptable results for Kt compared to those using MLP. Accuracy analysis

203

illustrates very accurate results for the RBF model when conditions lie within the range used for training the model. When extrapolating input parameters outside that range, although results seem less accurate than for interpolation cases, the RBF model still performs well and presents very acceptable outputs. The RBF model was assessed under larger scale conditions. Outputs under these conditions also showed good agreement. This shows that the performance of the model is not affected significantly by scale changes and the model has the potential to be used in real applications. The sensitivity analysis showed that the non-dimensional spatial parameters x/Lo and y/Lo were the most effective among all the parameters that influence the process. The non-dimensional hydraulic parameters ξo and Hi/h were also found to have a considerable influence on the results. Without these parameters as inputs to the network, the lowest correlation coefficient and the greatest error were obtained. It is concluded that the proposed model offers the potential of a design tool to predict spatial wave transmission coefficients behind submerged breakwaters. The results of this study can be applied not only for wave modeling behind submerged structures but also for flow models around these structures by importing the wave data into the available models and calculating flow patterns behind such structures. A plan for future work on ANN modeling of flow around submerged breakwaters is going to be conducted by the authors to provide a 3D flow design tool using both experimental data collected in the laboratory and artificial data provided by numerical models. Appendix A. A simplified design tool and initial prediction scheme A step by step procedure for practical applications is outlined in this appendix with the intention of providing an initial prediction scheme according to the proposed ANN model described in the paper. The main aim is to introduce a simplified tool for preliminary design purposes which can be used in submerged breakwater design. A 3D spatial Kt calculator is described below, based on the proposed non-dimensional RBF model. As explained in Section 5, the final optimized ANN model has one radial basis function layer with 25 nodes. Below, the simplified ANN model is presented in an easy to use and user friendly form and with a step by step procedure including details about the model mathematics. For more details and theoretical aspects see Haykin (1999) or Hagan et al. (1996). Although the model is calibrated and designed using regular wave data, similar prediction can be adopted using Significant Wave Height (Hs) or Zero Moment Wave Height (Hmo) for incident wave height and the peak wave period, Tp for wave period as inputs. However, some considerations might be required in applying regular wave results to random sea‐states. Referring first to the non-dimensional parameters used throughout the paper, an input vector I should be considered: I ¼ ½ξ0 ; Hi =h; hs =Hi ; B=Hi ; x=Lo ; y=Lo 

ðA:1Þ

and which is re-scaled in the range [0; 1], such that: In ¼ ðI−I min Þ=ðI max −I min Þ

ðA:2Þ

where vectors Imin and Imax are characterized by the values: I min ¼ ½3:5498; 0:0333; 0:0000; 1:7220; −0:2868; −0:2868

ðA:3Þ

I max ¼ ½14:6778; 0:6040; 11:2500; 19:500; 0:5429; 1:7825:

ðA:4Þ

Considering the center and weight matrices defined below, the input signals are passed through the network from the input layer to the hidden layer and from hidden layer to output layer respectively. The seventh column of the center matrix c is the vector of bias

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neurons for the hidden layer (b1), which is used to adjust the activation value in Eq. (A.7). 8 9 0:4598 0:0526 0:5263 0:4804 0:1345 0:0438 0:8326 > > > > > > > 0:4383 0:0501 0:1746 0:7012 0:1358 0:1980 0:8326 > > > > > > > > > 0:4597 0:0175 0:7692 0:7469 0:9259 0:7327 0:8326 > > > > > > > > 0:4509 0:0659 0:0000 0:7278 0:0247 0:1386 0:8326 > > > > > > > > 0:4231 0:0419 0:3726 0:6695 0:1852 0:7723 0:8326 > > > > > > > > 0:4509 0:0659 0:0000 0:7278 0:0247 0:2624 0:8326 > > > > > > 0:4997 0:0621 0:3102 0:5411 0:1177 0:0438 0:8326 > > > > > > > > 0:6397 0:0891 0:1285 0:4902 0:1927 0:3659 0:8326 > > > > > > > > > > 0:0272 0:5562 0:0000 0:0699 0:0000 0:0000 0:8326 > > > > > > > > > > 0:3344 0:5391 0:0000 0:0747 0:5133 0:3202 0:8326 > > > > > > > > 0:2451 0:5427 0:0622 0:0310 0:3778 0:2473 0:8326 > > > > > > > < 0:6682 0:0651 0:3027 0:5256 0:6120 0:3113 0:8326 > = c ¼ 0:5853 0:0643 0:4762 0:4255 0:2720 0:4113 0:8326 > > > > > > 0:9914 0:0000 1:0000 1:0000 0:2040 0:0750 0:8326 > > > > > > > 0:1493 0:5598 0:0306 0:0432 0:2493 0:1659 0:8326 > > > > > > > > 0:0000 1:0000 0:0000 0:0000 0:1092 0:0438 0:8326 > > > > > > > > > 0:4598 0:0526 0:5263 0:4804 0:3541 0:4637 0:8326 > > > > 0:2185 0:4971 0:0000 0:0877 0:3400 0:2522 0:8326 > > > > > > > > > > 0:3088 0:0584 0:5000 0:4516 0:3580 0:2574 0:8326 > > > > > > > > > > 0:3181 0:1231 0:0000 0:4680 0:1975 0:5297 0:8326 > > > > > > > 0:4597 0:0175 0:7692 0:7469 0:0617 0:2574 0:8326 > > > > > > > > > 0:4383 0:0501 0:1746 0:7012 0:1358 0:5050 0:8326 > > > > > > 0:9914 0:0000 1:0000 1:0000 0:3513 0:0750 0:8326 > > > > > > > > > > 0:4509 0:0659 0:0000 0:7278 0:0247 0:3861 0:8326 > > > > : ; 0:4509 0:0659 0:0000 0:7278 0:0123 0:6436 0:8326

w¼

8 −55:1298; −33:7780; −2:7259; 538:7780; −11:2170; −1444:5606; > > > > > > < 82:9272; −10:0918; −1:7088; 2:8396; 22:9205; 0:5419; −20:9583; > > −4:5373; 31:8786; −3:5441; 35:4056; −58:5264; −3:4593; 9:9345; > > > > : 11:2033; −4:9998; 4:0406; 1086:8719; −173:9348g:

K t ¼ ðO max −O min Þ:a2 þ O min

References

ðA:5Þ

ðA:6Þ

ðA:7Þ

where ao ð7; 1Þ ¼ 1

ðA:8Þ

and constant value d is: d ¼ sqrtð− logð:5ÞÞ=sp

ðA:9Þ

where sp = 1.0. The Euclidean distance between the network input vector ao and RBF unit's center vector c, is defined as follows: distðci −ao ;i Þ ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi X ci −ao;i

for i : 1…7

ðA:10Þ

where ci and ao,i are the coordinates of c and ao,i in dimension i. Whereas a linear activation function has been applied between the hidden and the output layers, the scaled model output can be computed as follows: a2 ¼ w:a1 þ b2

ðA:12Þ

where Omin = 0.0844 and Omax = 1.0742.

The outputs of the input layer are calculated based on the Euclidean distance between the network input vector and RBF unit's center vector c. The hidden layer is a linear combination of basis functions. Activation functions of radial basis type are used to determine the values of hidden neurons, such as:  2 a1 ði; 1Þ ¼ exp −ðdistðcði; :Þ; ao Þ:dÞ ; for i : 1…25

Finally, the model output needs to be rescaled to obtain the estimate of the transmission coefficient Kt:

ðA:11Þ

where b2 = 0.4411 is the bias neuron for the output layer; this is used to adjust the activation value in Eq. (A.11).

Bellotti, G., 2004. A simplified model of rip currents systems around discontinuous submerged barriers. Coast. Eng. 51 (4), 323–335. Broomhead, D., Lowe, D., 1988. Multivariable functional interpolation and adaptive networks. Compl.Syst. 2, 321–355. Buccino, M., Calabrese, M., 2007. Conceptual approach for prediction of wave transmission at low-crested breakwaters. J. Waterw. Port Coast. Ocean Eng. 133 (3), 213–224. Buccino, M., Vicinanza, D., Caceres, I., Calabrese, M., 2009. 3D wave field behind impermeable low crested structures. J. Coast. Res. 56, 477–481. Caceres, I., Stive, M.J.F., Sanchez-Arcilla, A., Trung, L.H., 2008. Quantification of changes in current intensities induced by wave overtopping around low crested structures. Coast. Eng. 55, 113–124. Calabrese, M., Vicinanza, V., Buccino, M., 2002. Large scale experiments on the behaviour of low crested and submerged breakwaters in presence of broken waves. Proc. 28th Int. Conf. on Coastal Engineering. ASCE, Cardiff, UK, pp. 1900–1912. d'Angremond, K., van der Meer, J.W., de Jong, R.J., 1996. Wave transmission at low-crested structures. Proceedings of the 25th International Conference on Coastal Engineering Orlando, FL, USA, ASCE, pp. 3305–3318. Draper, N.R., 1984. The Box–Wetz criterion versus R2. J. Stat.Soc. 147, 100–103. Goda, Y., Ahrens, J.P., 2008. New formulation of wave transmission over and through lowcrested structures. Proceedings of the 31st International Conference of Coastal Engineering, 4. Goda, Y., Takaiama, T., Suzuki, Y., 1978. Diffraction diagrams for directional random waves. Proceedings of the International Conference on Coastal Engineering ASCE, pp. 628–650. Hagan, M.T., Demuth, H.B., Beale, M.H., 1996. Neural Network Design. PWS Publication. Haller, M.C., Darlymple, R.A., Svendsen, I.A., 2002. Experimental study of nearshore dynamics on a barred beach with rip channels. J. Geophys. Res. 107 (C6), 14-1–14-21. Haykin, S., 1999. Neural Networks: A Comprehensive Foundation, 2nd edn. Prentice Hall, New Jersey. Hur, D.S., Lee, W.D., Cho, W.C., 2012. Three-dimensional flow characteristics around permeable submerged breakwaters with open inlet. Ocean Eng. 44, 100–116. Johnson, H.K., Karambas, T.V., Avgeris, I., Zanuttigh, B., Gonzalez-Marco, D., Caceres, I., 2005. Modelling of waves and currents around submerged breakwaters. Coast. Eng. 52 (10–11), 949–969. Lamb, H., 1924. Hydrodynamics. Cambridge University Press. Losada, I.J., Silva, R., Losada, M.A., 1996. 3-D non-breaking regular wave interaction with submerged breakwaters. Coast. Eng. 28, 229–248. Mase, H., Sakamoto, M., Sakai, T., 1995. Neural network for stability analysis of rubblemound breakwaters. J. Waterw. Port Coast. Ocean Eng. 121 (6), 294–299. McGill, R., Tukey, J.W., Larsen, W.A., 1978. Variations of boxplots. Am. Stat. 32, 12–16. Medina, J.R., 1999. Neural network modelling of runup and overtopping. ASCE Proc Coastal Structures Santander, 1 421–429. Panizzo, A., Briganti, R., 2007. Analysis of wave transmission behind low-crested breakwaters using neural networks. Coast. Eng. 54 (9), 643–656. Park, J., Sandberg, I.W., 1993. Approximation and radial-basis function networks. Neural Comput. 5, 305–316. Penny, W.G., Price, A.T., 1952. The diffraction theory of sea waves and the shelter afforded by breakwaters. Philos. Trans. R. Soc. Lond. A 244, 236–253. Poggio, T., Girosi, F., 1990. Networks for approximation and learning. Proc. IEEE 78, 1481–1497. Robert, J., Howlett, L.C.J., 2001. Radial Basis Function Networks 2: New Advances in Design. Physica-Verlag, Herdelberg, Germany. Rumelhart, D.E., Hinton, G.E., Williams, R.J., 1986. Learning representations by backpropagating errors. 323 (6088), 533–536. Seabrook, S.R., Hall, K.R., 1998. Wave transmission at submerged rubble mound breakwaters. Proceedings of the 26th Int. Conf. on Coastal Engineering, ASCE, pp. 2000–2013. Stein, R., 1993. Selecting data for neural networks. AI Expert. 8 (2), 42–47. Tukey, J.W., 1977. Box-and-whisker plots. Explanatory Data Analysis. Addison-Wesley, Reading, MA, pp. 39–43. van der Meer, J.W., Briganti, R., Zanuttigh, B., Wang, B., 2005. Wave transmission and reflection at low-crested structures: design formulae, oblique wave attack and spectral change. Coast. Eng. 52 (10–11), 915–929. van Gent, M.R.A., van den Boogaard, H.F.P., 1998. Neural network modelling of forces on vertical structures. ASCE, Proc. of the Coastal Engineering Conference, vol 2, pp. 2096–2109. van Gent, M.R.A., van den Boogaard, H.F.P., Pozueta, B., Medina, J.R., 2007. Neural network modelling of wave overtopping at coastal structures. Coast. Eng. 54 (8), 586–593. van Oosten, R., Peixó Marco, J., van der Meer, J.W., Verhagen, H.J., 2006. Wave transmission at low-crested structures using neural networks. ASCE, ICCE, San Diego 4932–4944. Vicinanza, D., Caceres, I., Buccino, M., Gironella, X., Calabrese, M., 2009. Wave disturbance behind low-crested structures: diffraction and overtopping effects. Coast. Eng. 56 (11-12), 1173–1185. Werbos, P.J., 1988. Backpropagation: past and future. Neural Networks. IEEE International Conference on. Neural Networks, New York, vol. 1, pp. 343–353. Wilmott, C.J., 1981. On the validation of models. Prog. Phys. Geogr. 2, 184–194.