Prediction of non-breaking wave induced scour depth at the trunk section of breakwaters using Genetic Programming and Artificial Neural Networks
Introduction
Coastal structures such as breakwaters are constructed to protect harbors and vessels from wave attacks. Proper and optimum initial design of these structures can eliminate the main construction problems, such as the structure instability, which could cause significant unforeseen expenditure. Therefore, optimizing the design of coastal structures is fundamental.
Scour, which may act as a threat to the stability and functionality of marine structures, is one of the main reasons for the failure of coastal [11], [17], [23], [31] and offshore (e.g. [21], [22] structures. Therefore, protecting structures against scour is critical in the construction of well-functioning man made harbors. To do this, the accurate prediction of maximum scour depth at coastal structures has inevitable importance. Although several studies have been conducted on scour at coastal structures, the complexity of onshore hydrodynamic and complex interaction between incoming waves, bed sediments and structure has impeded the accurate maximum scour depth prediction. Scour at breakwaters or seawalls (vertical or inclined) can be categorized into two main classes: scour at the head of coastal structures; and scour at the trunk section of coastal structures (due to breaking or non-breaking waves). Since the present paper focuses on predicting of the maximum scour depth at breakwaters due to non-breaking waves (hereafter ), only the non-breaking wave-induced scour depth at the trunk section of coastal structures has been discussed here. It is noted that is the ultimate value of scour depth when the equilibrium bottom profile is reached and it is independent of time.
Scour at inclined and vertical breakwaters due to non-breaking waves was investigated in several studies based on small-scale experiments. Sawaragi [28] and Baquerizo and Losada [5] investigated the relation between the wave reflection and the equilibrium scour depth at a rubble-mound breakwater and suggested that the scour depth becomes larger with the increase of the reflection coefficient (). Similarly, using small-scale experiments, Oumeraci [23] studied the effect of breakwater slope on and suggested that the maximum scour depth in front of a vertical breakwater is larger than that at sloped breakwaters. Furthermore, he indicated that the key mechanism for scour due to non-breaking waves is the action of standing waves (fully or partially), which leads to a steady streaming pattern. Carter et al. [6] investigated the regular and irregular wave-induced scour depth at vertical breakwaters, and showed that the scour and deposition pattern in front of the vertical breakwaters emerges in the form of alternating scour and deposition developing parallel to the shoreline. This finding has also been obtained by Baquerizo and Losada [5].
Soft computing approaches like Artificial Neural Networks (ANNs) and Genetic Programming (GP) have been successfully employed for the prediction of scour depth in various fields of coastal engineering, such as the estimation of scour depth below free overfall spillways [27], the estimation of scour around submarine pipelines [14], the prediction of scour depth under live-bed conditions at river confluences [4], the prediction of scour depth in bridges [1], the prediction of scour at a bridge abutment [2], the determination of the most important parameters on scour at coastal structures [33]; [25], the study of scour below submerged pipeline [3]. Regarding the mentioned studies, GP and ANNs can predict scour depth at coastal structures with high precision, and, to the best knowledge of the authors, these approaches have not been implemented in the prediction of the . Therefore, ANNs and GP have been used in this study as robust and promising tools. Furthermore, GP is capable of producing physically-sound and accurate solutions in the form of mathematical equations. Using this capability of GP, a new formula was developed for the prediction of .
This study is structured as follows: Section 2 shows the overview of scour governing parameters; Section 3 presents ANNs and GP concepts. The modeling approach and the data at the basis of the analyses are reported in Section 3; the results and discussions are given in Section 4; the sensitivity analysis is given in Section 5 and finally Section 6 contains this study summary and the conclusion.
Section snippets
Scour governing variables and formulas
Scour at the trunk section of breakwaters due to non-breaking waves depends on three classes of parameters: the wave characteristics, the sediment properties and the breakwater configuration. Several small-scale experimental studies are available that provide useful information about the governing parameters of scour at breakwaters. Among the most important experimental studies that also led to empirical formulas to predict scour depth we find the following.
Xie [32] examined the scouring
Genetic Programming (GP)
GP is borrowed from the process of evolution occurring in the nature (survival of the fittest). GP employs a “parse tree” structure for the search of its solutions, which are continuingly evolving and never fixed. Unlike the most soft computing approaches, like ANNs, GP solutions are in the form of tree structure, mathematical equations or computer programs (see Fig. 2). Furthermore, there is no assumption made on the structure of the relationship between the independent and dependent
Datasets description
To evolve the GP model, a combination of [32], [29] and [16] published datasets was used. All such data (95 data points), as well as the related finding come from small-scale flume experiments. It is clear that all findings derived from the mentioned datasets, as it happens for all studies based on small-scale experiments, can be influenced by scale effects and their use for practical applications should be made with some caution. Further, being based on flume experiment (2D flow) all results
Evaluation of the existing formulas
Best known empirical formulas for the prediction of the maximum scour depth are those of [32], [29], and [16] (Section 1). In this section the performance of these formulas in the prediction of the maximum scour depth for the different datasets (Section 3.1) has been investigated on the basis of statistical error parameters. Fig. 6a, b, and c show that the performance of the empirical formulas of [32], [29], and [16] is fair in predicting the maximum scour depth for their own dataset. However,
Uncertainty and reliability assessment
After developing the final models with ANNs and GP, their predictions for other test data sets may be biased and it is possible that the models cannot perform adequately well for the available data spread over the entire domain of data set. To have more trustworthy models, a resampling technique, the K-fold cross validation, has been applied to the data set as a whole. In this technique, the whole data set is randomly partitioned into K equal-sized folds; K-1 folds are used for training and the
Sensitivity analysis
Sensitivity analysis is a conventional method for determining the relative significance of input parameters in the modeling process. Employing irrelevant or insignificant input parameters can lead to complex models, which are very difficult to evaluate and interpret. One of the major capabilities of genetic programming is its inherent power in the determination of the variables importance in the evolved model, in a way that the unimportant variables are gradually omitted in the final evolved
Summary and conclusion
In this study, the non-breaking wave-induced scour depth at the trunk section of breakwaters has been studied by Genetic Programming (GP) and Artificial Neural Networks (ANNs) methodologies. Experimental data sets collected from the available literature have been used for developing the models. The developed models predict the relative scour depth (Smax/Lnb) as function of the reflection coefficient (Cr), the non-breaking wave steepness (Hnb/Lnb), the Shields parameter (θ), and the relative
Acknowledgments
The authors are grateful to Professor William Allsop and Professor Marcel R.A. van Gent for their constructive comments. Also, the authors express their sincere gratitude to Professors M. Sumer, J. Fredsøe, K.H. Lee, N. Mizutani, S.L. Xie for providing their valuable experimental data for this paper.
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