Elsevier

Coastal Engineering

Volume 152, October 2019, 103529
Coastal Engineering

A Genetic Programming based formula for wave overtopping by crown walls and bullnoses

https://doi.org/10.1016/j.coastaleng.2019.103529Get rights and content

Highlights

  • A new formula representing the reducing effects of crown walls and bullnoses on the overtopping discharge is proposed.

  • The formula is developed based on Genetic Programming fitting up to 1 000 new and existing data.

  • The formula applies to smooth structures under breaking and non-breaking waves.

  • Different combinations of structural elements can be represented by means of the same formulation.

  • The formula provides accurate, conservative and physically meaningful estimates of the overtopping discharge.

Abstract

The purpose of this contribution is to propose a new method for the parametrization of the reductive effects induced by crown walls and bullnoses on the average wave overtopping discharge (q) at coastal structures. The method consists of a formula for calculating an influence factor γ*GP to account for the single or combined effects of the structural elements. The formula for γ*GP is conceived to be included in the q formula by EurOtop (2018). The new formula was developed on the basis of the Genetic Programming (GP) technique trained on a database of nearly 1 000 data on wave overtopping at dikes with berms or promenades, crown walls and bullnoses. Part of the data are derived from new experiments carried out by the authors to extend the experience available from the literature and create a database of structure configurations sufficiently wide and appropriately assorted to be used for training the GP. The rough formula for predicting γ*GP obtained by the pure application of the GP was optimized to achieve a greater accuracy in the representation of both the breaking and non-breaking wave conditions. The estimations of q obtained with the new influence factor γ*GP are physically meaningful and satisfactory accurate, and overcome the underestimation bias affecting the predictions from the available formulae.

Introduction

The intensification of storm events, in combination with the sea level rise, exposes the traditional costal protections, such as dikes and seawalls, to increasing wave loads and overtopping rates. Several measures and solutions to mitigate coastal risks have been investigated (inter alia, Touili et al., 2014; Zanuttigh et al., 2014). One of the engineering-based solutions consists in the upgrade of existing defense structures by adding structural elements such berms, armour and crest units, crown walls, etc. (Burcharth et al., 2018). The addition of crown walls or the inclusion of bullnoses and parapets represent economic and aesthetic-compatible solutions to effectively reduce the overtopping discharges (q), and their employment is increasing rapidly.

Up to the 2010's, the literature studies on the effects of wave walls, parapets and bullnoses on the reduction of q are few and relatively fragmented (EurOtop, 2007). Among these, it is worthy to mention the earliest experiments on recurved parapets on vertical and sloping seawalls carried out by Owen and Steele (1991), the study by Kortenhaus et al. (2001) on the influence of overhanging deflectors on top of vertical walls or steep embankments and the experience collected within the FP7 EU-project CLASH on oblique and recurved parapets (Kortenhaus et al., 2003). The first systematic work on this topic is represented by the campaign of experimental investigations conducted by Van Doorslaer et al. (2015), VD, hereinafter. The study examined the typical defense structures of the Belgian coasts, characterized by smooth dikes, with a long and mildly sloping promenade above the still water level. The authors hypothesized to upgrade the existing structures by constructing a crown wall directly on the slopes or at the end of the promenades, including or not a bullnose, bn hereinafter. VD combined different geometries, collecting more than 1000 experiments on wave overtopping. The study prompted the first organic set of formulae parametrizing the combined effect of wall, bn, and promenades into a reduction coefficient γ* to be included in the EurOtop (2007) equations for the prediction of q. The expressions for γ* vary according to the combination of structure elements and their validity is limited to non-breaking wave conditions exclusively. Based on the results by Van Doorslaer et al. (2015), the coefficient γ* was adopted in the updated version of the EurOtop (2018) manual and applied to the new formula by Van der Meer and Bruce (2014) for the prediction of q, which replaces EurOtop (2007).

Recently, Zanuttigh and Formentin (2018), on the basis of a new set of experimental and numerical investigations, proposed a correction to the factor γ* in order to extend the work by VD to breaking wave conditions and to more conventional structure types, such as dikes with berms of limited width.

Both EurOtop (2018) and Zanuttigh and Formentin (2018) showed how the use of γ* apparently provides a physically-coherent representation of the wave overtopping at structures with walls and bullnoses, bns hereinafter. Both the works are based on the assumption that the original γ* factor conceived by VD for EurOtop (2007) could be straightforwardly applied to the EurOtop (2018) version of the equations for q, but never checked the consistency among predictions and measured values of q.

The present study starts with the revision and verification of the original and updated formulae for γ*, by re-applying the two methods to new and existing data. By comparing predictions and measurements of q, it was found that both the methods significantly underestimate the q-values.

The aim of this work is to present a completely new formula for the parametrization of the effects of crown walls and bns, which can ensure accurate and cautious estimates of q and, at the same time, which is of simple and straightforward application. The new method, which is meant to be directly applicable to the EurOtop (2018) formulae, was developed based on the Genetic Programming (GP). This innovative technique belongs to the field of the Artificial Intelligence and was already and successfully applied in Coastal Engineering for the prediction of the scour at the trunk section of breakwaters (Pourzangbar et al., 2017) and of the wave run-up (Power et al., 2019). Differently from more conventional machine-learning tools, such as neural networks, the predicting tools delivered by GP are algebraic formulae – and not black-boxes – which offer the tool developer the possibility of a direct interaction. Furthermore, the derived formula can be updated to fit new data. The learning phase of GP is extremely quick (a few minutes) and the computational effort significantly low, even in case of large amounts of data. GP is applied to the representation of the wave overtopping for the first time in this contribution.

The paper is organized as follows. Section 2 presents the new experimental and numerical database of wave overtopping experiments at dikes with berms, crown walls and bns conducted by the authors and used to verify the existing methods and calibrate the new one. The literature overview and the results of the application of the existing methods to the prediction of the overtopping discharges are given in Section 3. Section 4 firstly introduces the reader to the basic elements of the GP technique, and it then illustrates the scheme and the parameters adopted to apply the GP to the modelling of the effects of crown walls and bns. Finally, the Section presents the new formula for the prediction of the coefficient γ*, which was obtained by the application of the GP technique and conveniently adjusted to fit all the available data and to offer a simpler and more physically-based expression. The application of the GP-based formula to the available data is presented in Section 5. The conclusions of the work are finally drawn in Section 6.

Section snippets

The new database

A new campaign of laboratory experiments and numerical modelling on wave overtopping at structures with crown walls and bns was conducted. The campaign was aimed at extending the experience collected by VD considering a different structure type under breaking and non-breaking waves. For all the tested conditions, the wave breaking/non-breaking is supposed to occur for the combination of wave steepness and structure slope, i.e. in relation to the values of the Iribarren-Battjes breaker parameter

Application of the existing methods

This Section proposes the comparison of the new and existing data (i.e. the UB and the VD data) to the methods available from the literature (briefly recalled in Sub-section 3.1), in order to contemporary check the reliability of the data (Sub-section 3.2) and the adequacy of the formulae (Sub-section 3.3).

Results and verification of the new formula

This Section presents the results of the application of the GP method to the new database of numerical and experimental data (Sub-section 5.1) and to the database by VD (Sub-section 5.2).

For each dataset, the predictions of qGP were compared to the corresponding measurements (qmeas) and to the predictions obtained with the literature methods (qVD by Eq. (1) and qZF by Eqs. (6a) and (6b)). The performance of the new formula is quantitatively assessed and compared to the performance of the

Conclusions

This contribution proposed a new coefficient γ*GP to represent the effects of crown walls and bullnoses (bns) on the reduction of q. The coefficient is conceived to be included in the EurOtop (2018) formulae for the prediction q: it is supposed to replace the coefficient γ* in the non-breaking waves formula and the coefficients γv and γb in the formula for breaking waves.

The new formula for calculating γ*GP has been developed by applying the Genetic Programming (GP) technique to a set of

Acknowledgments

The authors would like to express their sincere gratitude to Dr. Koen van Doorslaer, for providing the experimental data, and to Dr. Jentsje W. van der Meer for the long-term cooperation.

The support of the European Commission through the Horizon 2020 project BRIGAID (“BRIdging the GAp for Innovations in Disaster resilience”, www.brigaid.eu) is gratefully acknowledged.

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