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An approach to optimize bed sill design using multi-objective evolutionary algorithms

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Abstract

Bed sills are a kind of grade control structure built at the same level as the river bed to protect it against erosion and to reduce the longitudinal slope of the channel. The aim of the present research was to determine the optimal dimensions of the structure by devising a new methodology to minimize cost and maximize stability, based on the optimization model of the multi-objective non-dominated sorting genetic algorithm (NSGA-II). This is achieved by calculating the width, height and distance between bed sills with these stability and cost objectives in mind. The NSGA-II algorithm is applied to solve the problem, and a set of Pareto points is obtained. The results are verified by considering the bed sills of the River Madarsoo in Iran and the River Plima in Italy, and comparing the results with those obtained using the Branch-and-Bound (B-and-B) algorithm. The results indicate that the cost and stability could respectively be improved by up to 18 and 20% in the River Madarsoo, and up to 18 and 26% in the River Plima, if the NSGA-II algorithm is used.

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Data availability

Some or all data, models, or code generated or used during the study are available from the corresponding author by request.

Abbreviations

\({\text{cost}} ({\text{x}} )\) :

Cost function

\({\text{D}} ({\text{j}} )\) :

Crowding distance of jth individual

\({\text{D}}_{m}\) :

Mean grain size of sediments

\({\text{D}}_{x}\) :

Grain size such that \({\text{x}}\)% (in weight) of grains are smaller

\({\text{d}}_{i}\) :

Average depth at bankfull discharge in incised reach

\({\text{F}}_{{\text{bo}}}\) :

Blanch's zero bed factor

\({\text{FS}}\) :

Safety factor

\({\text{F}} (x_{i} )\) :

Individual of population

\({\text{f}}_{{bed_{{}} sill}}\) :

Bed sill weight

\({\text{f}}_{\text{k}} (x_{\text{i}} )\) :

Objective function

\({\text{f}}_{i}\) :

Value of the i \(i\) th objective function

\({\text{f}}_{i}^{max}\) :

Maximum or ideal value of ith objective function

\({\text{f}}_{i}^{min}\) :

The minimum or the worst values for the ith objective function

\({\text{f}}_{i}^{j + 1}\) :

Ith objective function value of j+1th individual

\({\text{f}}_{i}^{j - 1}\) :

Ith objective function value of j−1th individual

\({\text{f}}_{u}\) :

Water uplift

\((f_{us} )_{h}\) :

Soil horizontal component in bed sill upstream

\((f_{us} )_{v}\) :

Soil vertical component in bed sill upstream

\(f_{uw}\) :

Water horizontal load in bed sill upstream

\({\text{f}}_{w}\) :

Water vertical load

\({\text{f}}_{x}\) :

Objective function

\({\text{G}}_{s}\) :

Specific Gravity of Soil Solids

\(G_{s} - 1\) :

Relative submerged density of the sediment

\(g\) :

Acceleration due to gravity

\(H_{s}\) :

Critical specific flow energy

\({\text{h}}_{b}\) :

Bed sill height with respect to scour

\({\text{h}}_{w}\) :

Water depth at uniform flow conditions

\({\text{K}}_{a}\) :

Coefficient of active earth pressure

\({\text{L}}\) :

Longitudinal distance between bed sills

\({\text{L}}_{p}\) :

Length of the reach

\(m\) :

Number of objectives

\(m_{1}\) :

Ratio of cut cost to construction cost

\(m_{2}\) :

Ratio of fill cost to construction cost

\({\text{n}}\) :

Number of bed sills

\({\text{n}}_{b}\) :

Bed sill porosity

\({\text{n}}_{p}\) :

Soil porosity

\(n_{s}\) :

Manning's roughness coefficient

\({\text{p}}_{t}\) :

Parent population

\({\text{Q}}_{t}\) :

Offspring population

\(q_{f}\) :

Design flood discharge per unit river width

\({\text{q}}_{i}\) :

Bankfull discharge in incised reach per unit river width

\(SI\) :

Sorting index of bed particles

\({\text{s}}_{0}\) :

Initial bed slope

\({\text{s}}_{e}\) :

Equilibrium bed slope of river

\({\text{Stability}} (x)\) :

Stability function

\({\text{UPRC}}_{\text{b}}\) :

Cost per unit volume

\({\text{V}}_{i}\) :

Bed sill volume

\({\text{V}}^{\prime}_{i}\) :

Cut volume

\({\text{V}}^{\prime\prime}_{i}\) :

Fill volume

\({\text{W}}_{\text{b}}\) :

Bed sill width

\({\text{W}}_{i}\) :

Weight or importance of the ith objective

\({\text{W}}_{r}\) :

River width

\({\text{y}}_{d}\) :

Bed sill height

\({\text{y}}_{g}\) :

General scour depth

\({\text{y}}_{s}\) :

Local scour depth with respect to the crest elevation

\({\text{y}}_{t}\) :

Drop to be removed from reach

\({\text{Z}}\) :

Scour coefficient

\(\gamma_{s}\) :

Soil unit weight

\(\gamma_{satb}\) :

Bed sill saturated unit weight

\(\gamma_{sub}\) :

Soil submerged unit weight

\(\gamma_{w}\) :

Water unit weight

\(\theta_{c}\) :

Critical shields' mobility parameter

\(\Sigma {\text{f}}_{\text{h}}\) :

Resultant horizontal force

\(\Sigma {\text{f}}_{\text{v}}\) :

Resultant vertical force

\(\Sigma {\text{m}}_{o}\) :

Sum of the overturning moments

\(\Sigma {\text{m}}_{r}\) :

Sum of the stabilizing moments

\(\varphi\) :

Soil angle of friction

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Authors and Affiliations

  1. Faculty of Civil, Water and Environmental Engineering, University of Shahid Beheshti, Tehran, Iran

    Mohammad Reza Majdzadeh Tabatabai, Saeedeh Adineh, Saeed Alimohammadi & Seyed Hosein Ghoreishi

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  1. Mohammad Reza Majdzadeh Tabatabai
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  2. Saeedeh Adineh
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  3. Saeed Alimohammadi
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  4. Seyed Hosein Ghoreishi
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Correspondence to Mohammad Reza Majdzadeh Tabatabai.

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Tabatabai, M.R.M., Adineh, S., Alimohammadi, S. et al. An approach to optimize bed sill design using multi-objective evolutionary algorithms. Environ Earth Sci 80, 480 (2021). https://doi.org/10.1007/s12665-021-09688-2

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