ABSTRACT
Breakwaters are used to reduce incoming wave energy at harbors and shorelines. This paper presents a comparison of novel two-dimensional hybrid intelligent models for the idealization of the effects of waves on the performance of moored rectangular floating breakwaters (FBs). Fluid structure interactions (FSIs) were idealized by airy-type monochromatic regular waves generated in a numerical wave tank. The coupled Volume of Fluid-Fast Fictitious Domain (VOF-FFD) interpolation method was used to evaluate FB motions. Different forms of Least Squares Support Vector Machine Methods (LSSVMs) that utilized 183 data streams were used to model FB performance for different wave height-to-water depth ratios, dimensional aspect ratios, and specific length-to-water depth ratios. Of those, 80% were used to train the model and 20% to test it. Parametric studies have shown that during training a Least Squares Support Vector Machine Method-Bat Algorithm (LSSVM-BA) with R2 = 0.8725, MAE = 0.0276, and RMSE = 0.0488 presents the most appropriate model for the evaluation of FB performance. Notwithstanding this, during testing a Least Squares Support Vector Machine Method-Cuckoo Search (LSSVM-CS) Algorithm with corresponding values of 0.6841, 0.0519, and 0.0708 performs better.
HIGHLIGHTS
Hybrid intelligent methods can effectively predict floating breakwater wave transmission coefficients.
Novel two-dimensional hybrid models were used to model floating breakwater performance.
LSSVM-BA and LSSVM-CS models performed best for training and testing, respectively.
INTRODUCTION
Breakwaters are used to reduce incoming wave energy at harbors and shorelines. Contrary to mount breakwaters, floating breakwaters (FBs) are preferred as they are less costly in terms of installation and dissipate wave energy without constraining under-water flow in areas with tidal variations (Rafic & Pascal 2009; Dai et al. 2018; Ghazvinian & Karami 2024). The fast and accurate estimation of FB wave transmission coefficients remains challenging primarily because of the lack of unified models and associated uncertainties. Traditionally, methods using deterministic empirical methods and experimental results have been used (Ghazvinian et al. 2020a; Ghazvinian & Karami 2023a,2023b, 2023c). For example, Zhang & Li (2014) suggested the use of empirical formulae based on modified Boussinesq wave equations for the fast evaluation of wave transmission coefficient on permeable rubble mounds and pile-type breakwaters. van der Meer & Daemen (1994) developed practical design formulas and graphs, based on extensive experimental data, to predict the transmission wave for statically stable submerged, low-crested, and dynamically stable FB. d'Angremond et al. (1996) introduced an expression for the transmission coefficient of a low-crested breakwater based on structural and wave parameters. Detailed laboratory tests to estimate transmission coefficients have also been carried out by Kramer et al. (2005), Calabrese et al. (2008), Peng et al. (2009), Hur et al. (2012), Melito & Melby (2002), and Laju et al. (2011). From an overall perspective comparisons of laboratory tests against empirical formulae show that the results of van der Meer & Daemen (1994) can be applied to rubble mound breakwater installations. On the other hand, when the equation by d'Angremond et al.(1996) is used, large deviations may occur in the case of permeable breakwaters with large transmission coefficients.
The hydrodynamic efficiency of FB may vary depending on their design specification and operational conditions. For example, Ji et al. (2015, 2016, 2017) compared linear hydrodynamic theory and experiments to show that a cylindrical FB (CFB) consisting of rigid cylinders and a flexible mesh cage has better hydrodynamic performance than double pontoon-type structures. Liu & Wang (2020) investigated the hydrodynamics of moored box-type FB with different cross-sections using the smoothed particle hydrodynamics (SPH) method. Their study accounted for key hydrodynamic parameters such as the breakwater density, immersion depth, ballast-water gravity and wave conditions and showed that FB performance may be significantly affected by the wave conditions and immersion depth. Ji et al. (2018) studied experimentally the hydrodynamic performance of double-row rectangular FB with porous plates. They carried out a series of two-dimensional experiments and concluded that double-row FB systems perform better to mitigate the transmission wave. Cho (2016) studied the influence of wave transmission on an innovative floating rectangular breakwater. Their results showed that unique design characteristics such as porosity and deep side plates may reduce the influence of hydrodynamic loads.
Intelligent and hybrid methods are computationally versatile and therefore useful for the evaluation of the hydrodynamic performance of an FB. Basser et al. (2014) compared the application of two support vector regression (SVR) FB types, namely polynomial-based (SVR_poly) and radial basis function (RBF)-based SVR (SVR_rbf) with adaptive neuro fuzzy system (ANFIS), and adaptive neural network (ANN) to estimate the best parameters for the design of a protective spur dike. The authors observed that the SVR achieves better accuracy in terms of percentage reduction in the scour depth with a smaller network size, compared to the ANFIS and ANN approaches. Also, the SVR_rbf model having values of 0.7200 and 0.1200, respectively, for R and RMSE (root mean square error) indicators is more accurate compared to the SVR_poly approach with R = 0.6210 and RMSE = 0.1500.
More recently, the work by Ehteram et al. (2018) demonstrated that genetic, particle swarm and especially shark algorithms can be efficient for optimization in reservoir operation and water supply. However, the complexity of such methods implies the need for further testing and validation. As an alternative Saghi et al. (2021) introduced a fast and efficient numerical model for the idealization of moored FB motions in regular and irregular waves. In their work, they assessed the hydrodynamic performance of an FB by the Cuckoo Search-Least Square Support Vector Machine model (CS-LSSVM) and demonstrated that a suitable combination of sidewall mooring angle and the aspect ratio of the FB could help attenuate incoming waves to a minimum height.
The review of previous studies shows that so far, few researches have been done in the field of using intelligent methods to idealize the effects of waves on the performance of moored rectangular FBs. This paper presents a rapid method for the prediction of the transmission coefficient of a rectangular FB. The numerical model of Saghi et al. (2021) which combines a fast fictitious domain with a volume of fluid (VOF) method to track free surface effects is used for motion prediction. A brief description of different hybrid intelligence models that were assessed and compared in this study is given in Section 2 alongside an explanation of the methods and statistical tools used to evaluate the various predictions. The results presented in Section 3 discuss the performance indicators through a comprehensive set of performance diagrams. Conclusions are given in Section 4.
MATERIALS AND METHODS
Governing equations and boundary conditions
(a) A real FB (Marine biotech, 2007) and (b) schematic sketch of NWT rectangular moored FB configuration.
(a) A real FB (Marine biotech, 2007) and (b) schematic sketch of NWT rectangular moored FB configuration.
In this figure, a moored floating breakwater with dimensions b0 and h0 is placed in a part of the wave tank such that the incoming wave height is equal to Hinc and the transmitted wave height is equal to Htra.
is the gradient operator, υ is the kinematic viscosity, and υt is the turbulent viscosity.
was used to calculate Kt as:
Model validation
Model validation in the generation of the airy wave by using a wave maker theory.
Model validation in the generation of the airy wave by using a wave maker theory.
Comparison of transmission coefficients estimated by the developed model and the experimental results of Cui et al. (2020).
Comparison of transmission coefficients estimated by the developed model and the experimental results of Cui et al. (2020).
Intelligent hybrid models
An LSSVM model (Ghazvinian et al. 2020b, 2021) was used to estimate the transmission coefficient of the rectangular floating breakwater. Various optimization algorithms namely the Bat Algorithm (BA), Firefly Algorithm (FA), Grasshopper Optimization Algorithm (GOA), and Cuckoo Search (CS) Algorithm were coupled to the LSSVM model to find the optimum parameters of the model. Input parameters comprised 183 big data streams including HI/Hw (the ratio of incident wave height-to-water depth), AR (breakwater aspect ratio), and 
(the ratio of the specific length of the breakwater to water depth) in different conditions. The number of input data streams in the training mode of the four proposed models was 146. This corresponds to 80% of the overall data streams made available. The remaining testing module comprised 37 data streams.
LSSVM model
The structure of the LSSVM model to predict transmission coefficient of a FB.
Based on Figure 4, the input and output parameters are defined as, 
and, y = kt, respectively.
Optimization algorithms
The BA is a new meta-innovative algorithm inspired by bat positioning behavior (Cheng 2010). This algorithm is based on the echolocation which is a sonar wave emitted by the microbats. It helps bats find prey and discriminate against the different kinds of obstacles or dangers on their way toward the prey in complete darkness (Srivastava & Sahana 2019).
A BA comprises the following three basic steps:
1. All bats can make a sound and receive it. Based on this ability, they can distinguish between food sources and barriers.
2. Bats fly randomly and while flying they have wavelength (λ), constant frequency (fmin) and velocity (Vi) in the Xi position. They can also generate sound pulses between 0 and 1.
3. The loudness of bats can vary from a large positive A0 value to a small positive Amin value.
is the new speed of bat, 
is the previous speed, 
the new position, 
the previous position and β is a random vector with arrays between 0 and 1. In the first step, a random number between fmin and fmax is assigned to each bat. The bat speed and position are then updated based on Equations (9) and (10). Then a random number is generated. If the pulse generation rate is less than this random number, the local search is performed using the production of a random step based on Equation (11):
The FA is a population-based and random optimization algorithm based on the behavior of fireflies in placental uptake as introduced by Yang (2009). The three basic hypotheses of this algorithm are:
(1) There is no specific gender for fireflies.
(2) Each firefly is absorbed by other fireflies according to their light intensity.
(3) In problems of maximization, the amount of light intensity is directly related to the objective function, and in problems of minimization, the intensity of light is inversely related to the objective function.
- (4) The movement of a firefly to another firefly is defined as:(12)
The CS algorithm is based on the cuckoo's deceptive behavior when laying its eggs in other birds' nests (Yang & Deb 2009). In general, this algorithm has three general rules:
(1) Each cuckoo lays an egg only once at a time and lays it in a randomly selected nest.
(2) The best nests and the highest quality eggs are passed on to the next generation.
(3) The number of host nests available is constant and cuckoo eggs will be identified by the host bird of probability Pa
Each of these selected algorithms has its own advantages. For example, the advantage of the BOA is the powerful combination between a population-based algorithm and the local search, however, it is more efficient for local searches (Heraguemi et al. 2015). On the other hand, an advantage of CS is that its global search uses L'evy flights or processes, rather than standard random walks. As L'evy flights have infinite mean and variance, CS can explore the search space more efficiently than algorithms by standard Gaussian processes. This advantage, combined with both local and search capabilities and guaranteed global convergence, makes CS very efficient (Yang & Deb 2014). The GA allows for the participation of all search agents. The algorithm avoids falling into the trap of local optimization and then convergence and creating a balance between global and local search capabilities (Saremi et al. 2017). Finally, the advantage of the firefly optimization algorithm is that it automatically subdivides data streams and can deal with multimodality (Yang & He 2013).
Coupling of LSSVM and optimization algorithms
(1) Determine the basic parameters of optimization algorithms.
(2) Divide laboratory data into two courses of training and testing.
(3) Produce the primary population.
(4) Effect min. square backup vector machine training according to training course data and the decision variables of the optimization algorithm.
(5) Test the least squares backup vector machine and determine the objective function for the optimization algorithm.
(6) Terminate the condition control and if it is observed, return the optimal values of the parameters of the least squares support vector machine. If this is not possible then update the position in the optimization algorithms and go back to step 4.
Flow chart of couple of LSSVM and optimization algorithms.
Statistical indicators
R2 indicates the proportion of the variance between the observed values and the predicted values from intelligent models. The closer it is to 1, the better the correlation between the observed data and the results of the intelligent model. The MAE and RMSE indices help identify the error band. The closer the value of these indices is to zero, the more accurate the answer predicted by the smart model (Dadrasajirlou et al. 2022). In Equations (14)–(16), term O represents the value obtained from the numerical model, P is the value predicted using the intelligent model, and N is the number of variables.
RESULTS AND DISCUSSION
Table 1 presents the results of evaluation indicators for five different models in training and test courses. The models compared were LSSVM (model 1), LSSVM-CS (model 2), LSSVM-FA (model 3), LSSVM-GOA (model 4), and LSSVM-BA (model 5). Results show that in the training course, model 5 with R2 = 0.8275 is superior to all other models. In this training period, Model 5 with MAE = 0.0276 and RMSE = 0.0488 is better performing. However, the value of the determination coefficient in model 2 (R2=0.6841) is higher than intelligent all other models. Thus, this model is more accurate. Within the testing period indices MAE and RMSE for model 2 are 0.0519 and 0.0708 and, therefore, lower than all other models. Thus, model 2 has the highest predictive power.
The results of evaluation indicators for different intelligent models
| Model number | Statistical parameter | R2 | MAE | RMSE | |||
|---|---|---|---|---|---|---|---|
| Model name | train | test | train | test | train | test | |
| 1 | LSSVM | 0.1061 | 0.1962 | 0.0926 | 0.0910 | 0.1111 | 0.1120 |
| 2 | LSSVM-CS | 0.7847 | 0.6841 | 0.0374 | 0.0519 | 0.0546 | 0.0708 |
| 3 | LSSVM-FA | 0.7849 | 0.6781 | 0.0365 | 0.0365 | 0.0714 | 0.0714 |
| 4 | LSSVM-GOA | 0.7848 | 0.6674 | 0.0361 | 0.0361 | 0.0727 | 0.0727 |
| 5 | LSSVM-BA | 0.8275 | 0.5989 | 0.0276 | 0.0276 | 0.0778 | 0.0778 |
| Model number | Statistical parameter | R2 | MAE | RMSE | |||
|---|---|---|---|---|---|---|---|
| Model name | train | test | train | test | train | test | |
| 1 | LSSVM | 0.1061 | 0.1962 | 0.0926 | 0.0910 | 0.1111 | 0.1120 |
| 2 | LSSVM-CS | 0.7847 | 0.6841 | 0.0374 | 0.0519 | 0.0546 | 0.0708 |
| 3 | LSSVM-FA | 0.7849 | 0.6781 | 0.0365 | 0.0365 | 0.0714 | 0.0714 |
| 4 | LSSVM-GOA | 0.7848 | 0.6674 | 0.0361 | 0.0361 | 0.0727 | 0.0727 |
| 5 | LSSVM-BA | 0.8275 | 0.5989 | 0.0276 | 0.0276 | 0.0778 | 0.0778 |
The estimated transmission coefficient of the models relative to the observed one (estimated using the developed model) in the training step: (a) model 1, (b) model 2, (c) model 3, (d) model 4, and (e) model 5.
The estimated transmission coefficient of the models relative to the observed one (estimated using the developed model) in the training step: (a) model 1, (b) model 2, (c) model 3, (d) model 4, and (e) model 5.
The estimated transmission coefficient of the models relative to the observed one (estimated by using the developed model) in the testing step: (a) model 1, (b) model 2, (c) model 3, (d) model 4, and (e) model 5.
The estimated transmission coefficient of the models relative to the observed one (estimated by using the developed model) in the testing step: (a) model 1, (b) model 2, (c) model 3, (d) model 4, and (e) model 5.
The prediction error distribution of the models in the training step: (a) model 1, (b) model 2, (c) model 3, (d) model 4, and (e) model 5.
The prediction error distribution of the models in the training step: (a) model 1, (b) model 2, (c) model 3, (d) model 4, and (e) model 5.
The prediction error distribution of the models in the testing step: (a) model 1, (b) model 2, (c) model 3, (d) model 4, and (e) model 5.
The prediction error distribution of the models in the testing step: (a) model 1, (b) model 2, (c) model 3, (d) model 4, and (e) model 5.
Previously, in the research of Hu et al. (2021), better performance of intelligent models was seen in the training course than in the test course.
Estimation of transmission coefficient of rectangular FBs with different AR and its error by using model 2
| Sample number | HI/Hw | AR |  | Kt (EXP.) | Kt (model 2) | Relative error |
|---|---|---|---|---|---|---|
| 1 | 0.067 | 2.560 | 0.416 | 0.55 | 0.5388 | −0.0203 |
| 2 | 0.033 | 2.250 | 0.500 | 0.50 | 0.5162 | 0.0324 |
| 3 | 0.033 | 4.000 | 0.417 | 0.48 | 0.4963 | 0.0339 |
| 4 | 0.025 | 3.240 | 0.625 | 0.72 | 0.6955 | −0.0340 |
| 5 | 0.025 | 1.440 | 0.624 | 0.68 | 0.6025 | −0.1139 |
| 6 | 0.025 | 1.000 | 0.625 | 0.62 | 0.6120 | −0.0129 |
| 7 | 0.025 | 1.960 | 0.626 | 0.78 | 0.7585 | −0.0275 |
| 8 | 0.025 | 4.710 | 0.749 | 0.60 | 0.6468 | 0.0780 |
| Sample number | HI/Hw | AR | | Kt (EXP.) | Kt (model 2) | Relative error |
|---|---|---|---|---|---|---|
| 1 | 0.067 | 2.560 | 0.416 | 0.55 | 0.5388 | −0.0203 |
| 2 | 0.033 | 2.250 | 0.500 | 0.50 | 0.5162 | 0.0324 |
| 3 | 0.033 | 4.000 | 0.417 | 0.48 | 0.4963 | 0.0339 |
| 4 | 0.025 | 3.240 | 0.625 | 0.72 | 0.6955 | −0.0340 |
| 5 | 0.025 | 1.440 | 0.624 | 0.68 | 0.6025 | −0.1139 |
| 6 | 0.025 | 1.000 | 0.625 | 0.62 | 0.6120 | −0.0129 |
| 7 | 0.025 | 1.960 | 0.626 | 0.78 | 0.7585 | −0.0275 |
| 8 | 0.025 | 4.710 | 0.749 | 0.60 | 0.6468 | 0.0780 |
Taylor diagrams of estimated values in the testing phase.
In the Taylor diagram (Taylor 2001) shown in Figure 10, the longitudinal distance from the origin of the coordinates represents the standard deviation, the radial lines represent the correlation coefficient, and the segmental lines represent the root mean square error values. By increasing the circle segment, the mentioned parameter value is increased. This means that during the testing phase models 2 and 1 have the highest and lowest accuracy, respectively.
CONCLUSIONS
This paper presented a comparison of novel two-dimensional hybrid intelligent models for the idealization of the effects of waves on the performance of moored rectangular FB. FSI modeling assumed airy and irregular waves generated in a numerical wave tank and a coupled VOF-FFD method was used to evaluate motions. From an overall perspective, results show that hybrid intelligent methods can be fast and practical in terms of predicting the values of FB wave transmission coefficients. A comparison of Kt for five hybrid intelligent models showed that model 5 (LSSVM-BA) performed the best with training error between −15 and 25%. However, during testing model 2 (LSSVM-CS) appeared to present a more appropriate error band in the range of −22 and 16%. It may therefore be concluded that whereas the training period can be used for preliminary algorithm evaluation, the most suitable phase to choose the best algorithm for FB performance optimization could be the test period. This is because new data are used during the test period and, therefore, the accuracy of each algorithm is determined at this stage.
Future works
The simultaneous use of intelligent and numerical methods can be very useful and efficient in increasing both the accuracy and the ability to quickly predict various hydrodynamic parameters. Therefore, this technique can be applied to the simulation of various marine structures, including other types of breakwaters.
FUNDING
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
CONFLICT OF INTEREST
The authors declare there is no conflict.
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