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Optimization of coastal structures: Application on detached breakwaters in ports
Ocean Engineering 63 (2013) 35–43
Contents lists available at SciVerse ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Optimization of coastal structures: Application on detached breakwaters in ports Ghassan Elchahal a,n, Rafic Younes b, Pascal Lafon c a
Institut Charles Delaunay, University of Technology of Troyes, Troyes, France Lebanese University, Rafic Harriri Campus, Beirut, Lebanon c Institut Charles Delaunay, University of Technology of Troyes, Troyes, France b
a r t i c l e i n f o
abstract
Article history: Received 20 December 2011 Accepted 12 January 2013 Available online 5 March 2013
An optimization approach using Genetic Algorithms (GA) has been developed to determine the optimal layout of detached breakwaters inports subject to wave disturbance and navigational constraints. It aims to modify the shape and location of detached breakwater to achieve an optimal solution. This approach constitutes an advanced step in the design of coastal structures compared to conventional methods. The detached breakwater is parameterized by the coordinates of its nodes which constitute the variables of the optimization problem. The initial population is randomly generated inside a defined search space to ensure large consideration of various potential solutions. Mutation with random generation has been included at each iteration to renew the worse individuals and enrich the GA with new solutions. One and two segment breakwaters have been considered and compared. This optimization approach has been performed for the Port of Beirut as a real engineering application for two cases: simplified and real bathymetry. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Layout of detached breakwaters Water wave mechanics Hydrodynamic modeling Shape optimization Genetic algorithms
1. Introduction The layout of detached breakwaters in ports, i.e. shape and location, is usually designed by assessing several concepts based on the designer engineering experience. The design has not been addressed as a hydrodynamic analysis problem where a solution can be obtained. This may result from several severe problems related to the difficulty of the hydrodynamic analysis and evaluation of such marine structures mainly due to their arbitrarily shapes. It is noticed that optimization applications related to hydrodynamic aspects of marine and coastal structures are rarely reported. However some research work on this subject has been carried out. For example, Akagi and Ito (1984) optimized the heave motion of a hydrodynamic transparent semi submersible using a quadratic programming technique, Kagemoto (1992) optimized the arrangement of vertical floating cylinders in waves, Clauss and Birk (1996) focused on hydrodynamic shape optimization for large offshore structures (oil platforms) based on nonlinear programming algorithms. Elchahal et al. (2009) addressed an optimization problem for floating breakwaters. Isebe et al. (2008a) considered shape optimization approach to tackle beach erosion problems. They addressed the shape optimization of n
Corresponding author. Tel.:þ 33 3251 5651; fax: þ 33 3257 15675. E-mail addresses: [email protected] (G. Elchahal), [email protected] (R. Younes), [email protected] (P. Lafon). 0029-8018/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2013.01.021
geotextile tubes placed on the seabed in order to change the sea bottom shape and subsequently reduce beach erosion effects. Isebe et al. (2008b) also worked on the optimization of defense structures to minimize free surface elevation for fixed bathymetry coastal areas. They determined the optimum dimensions of groynes that yield to minimum wave height. They have used a simplified hydrodynamic model by considering a fixed bathymetry and thus excluded the effects of wave refraction due to water depth variation. To our knowledge this is the first paper to consider an optimization approach for design of breakwaters inside ports with a real engineering application. Detached breakwater in ports are usually designed to attenuate waves from secondary (nondominant) wave directions or for port extension projects. The detached breakwater have numerous advantages, i.e. moving the breakwater seawards allows more space for accommodating splash due to wave overtopping behind the breakwater and this significantly lowers the effects of overtopping at the limit of the working area within the harbor. This can also mean that the crest level of the structure can be reduced. A detached breakwater is likely to be shorter than a shoreconnected one; also the volume of the material needed will be less because of the reduced length and hence significant cost savings can be made. But on the other side the cost of the construction maybe higher because waterborne equipment has to be used. The main objective of this paper is to develop an optimization approach for the design of detached breakwater in ports using
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Genetic Algorithms (GA), i.e. find the optimum layout of the detached breakwater considering wave disturbance inside the port and navigational constraints. The idea reverts to the fact that in such fluid–structure interaction problems where the flow is affected by the presence of a structure in the studied domain, then the structure shape and location can be modified to attain a targeted flow. The optimization approach is defined and elaborated in Section 2. It consists of an objective or cost function, defined by the minimum length and most feasible location of the detached breakwater that could satisfy the wave disturbance criteria and navigational constraints. Wave disturbance inside the port requires a numerical model for wave propagation inside the port. The hydrodynamic model for wave disturbance applied in the optimization problem is defined in Section 3. The Genetic Algorithm is presented in Section 4. Section 5 includes the optimization results for an application for Port of Beirut as a real application, considering two cases for one segment and two segment breakwaters. In addition, a constant water depth bathymetry and the actual bathymetry were considered for each case as well. Finally, recommendations for further developments of optimization algorithm in the marine applications are provided.
Fig. 1. Parameterization of the structure and domain definition. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
2. Optimization problem An optimization problem consists in the minimization of a function fAR, also called cost or objective function, corresponding to the physical criteria to optimize. This function depends on xi, design parameters defining the detached breakwater layout within the admissible set X. In the application presented here, a given set of parameters xi defines a new shape for the breakwater, and thus a new domain Oðxi Þ for the computation. Then by using the hydrodynamic model presented in Section 3, the layout of the detached breakwater is derived and subsequently the value of the cost function f. In addition the free surface elevation Z in O is computed to determine the wave disturbance at a desired location or area inside the port. 2.1. Parameterization of the structure The selection of the optimization parameters is important for the problem definition and shall account for the imposed constraints. It is assumed that no definite shape is preferred and thus the structure is parameterized using the coordinates of the extremity points of the detached breakwater connected by straight edges, as shown in Fig. 1. These nodes, two or three, are free to move in the computational domain to achieve the optimal breakwater layout. Therefore, the numbers of variables will be four or six accordingly, xi ¼[x1,y1,x2,y2] or, xi ¼ ½x1 ,y1 ,x2 ,y2 ,x3 ,y3 The optimization problem is assumed to be finite dimensional constrained minimization problem, which is symbolically expressed as
find a design variable vector xi; to minimize the objective function f(xi) and subjected to the n constraints Cn(xi) r0.
Fig. 2. Port of Beirut–Lebanon.
waves by a long breakwater whereas the Northern Quay is exposed to North and North-Eastern waves and shall be protected with a new detached breakwater. The bathymetry of the Port of Beirut—Lebanon is presented in Fig. 3. It is observed that the water depth varies from 30 m to 5 m inside the port, where it is around 40 m outside the port.
2.3. Objective function The objective is to determine an optimal layout of the detached breakwater, i.e. minimum length and optimum location to achieve an acceptable wave disturbance inside the port. Subsequently an optimum solution will minimize the breakwater costs. The objective function is related to the length of the breakwater and expressed in terms of the geometrical coordinates of the structure. One segment breakwater: f ob ðxi Þ ¼ Min½ðx2 x1 Þ2 þ ðy2 y1 Þ2 1=2
2.2. Bathymetry definition The Port of Beirut—Lebanon (Fig. 2) is located at the eastern coastline of the Mediterranean. The Port is protected from the
Two segments breakwater: ( ) ½ðx2 x1 Þ2 þðy2 y1 Þ2 1=2 f ob ðxi Þ ¼ Min þ½ðx3 x2 Þ2 þ ðy3 y2 Þ2 1=2
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A numerical wave model is required to study the propagation of waves from deep water toward the area of interest inside the port. The well known mild slope equation, derived by Berkhoff (1972), for linear wave propagation has been applied. It combines the resolution of the three main physical mechanisms producing wave transformation: diffraction, reflection and refraction. In general, the diffraction process is of primary importance at the tips of breakwaters or harbor entrances. As harbors are semiclosed wall surrounded areas, the reflection mechanism must also be taken into account. Furthermore, the refraction effect plays a secondary role at areas with a relatively flat bottom, but is very relevant when passing over dredged navigation channels. The mild slope equation is expressed in its elliptic form as follows:
rðCC g rfÞ þ k2 CC g f ¼ 0
Fig. 3. Bathymetry definition of the port.
where (x1,y1), (x2,y2) and (x3,y3) are the coordinates of the extremity nodes of the detached breakwater segments. 2.4. Wave disturbance constraint The wave disturbance in a port is the most significant criteria to be considered in the determining the layout of the facilities prior to construction. The Northern Quay is the most exposed and satisfying the wave disturbance at this desired location will ensure acceptable conditions at other areas in the port. This constraint is expressed in terms of the maximum surface elevation at a desired area in front of the Quay. The allowable wave height constraint is expressed as follows: C 1 ðxi Þ ¼ 2 maxðZðx,y,tÞÞ o a where x and y represent the coordinates of points in the computational domain, and a is the allowable value height at the desired location. A value of a ¼0.3 m has been considered. 2.5. Navigational constraint The location of the new detached breakwater should not have impacts on the navigation inside the port. The ships shall still have sufficient space to sail in and out. The distance between the extremity of the detached breakwater and the existing breakwater shall be greater than a defined distance Dn (Fig. 5). A value of Dn¼ 100 m has been considered in the optimization approach. This constraint is included in the definition of the domain of optimal solutions detailed in Section 4.
where f is the velocity potential, C and Cg are phase and group velocity, respectively. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ð1 þ ð2kd=sinh 2kdÞÞ tanh kd and C g ¼ C C¼ k 2 Solving Berkoff’s mild slope equation allows obtaining the instantaneous velocity potential inside the fluid domain. Then, the free surface elevation with respect to the mean water level at each point within the harbor is easily computed, by the following relation:
Zðx,y,tÞ ¼
io fðx,y,tÞ g
The Berkoff equation is associated with the following boundary conditions related to reflection and radiation phenomena. 3.1. Reflection condition The breakwaters and the Quay walls represent rigid structures. Hence, the flow is reflected on these structures. The fully reflective boundary condition is really inappropriate since in reality partial reflection at the boundaries within a port invariably occurs. Moreover, reflecting boundaries are not always vertical or are fully reflecting. In order to treat these boundaries in the present model, they are schematized as partially reflective and introduced by using a mixed boundary condition instead of the full reflection condition in the manner proposed by Berkhoff (1976). @f kr 1 ¼ ik f, kr þ1 @n where kr represents the reflection coefficient (kr ¼ 0.3) The reflection condition is applied for all the structures marked in blue lines in Fig. 1. 3.2. Radiation condition
3. Hydrodynamic modeling Fluid is assumed to be ideal, flow is considered as irrotational and thus the linear wave theory can be applied. An incident monochromatic wave is characterized by its height H, its wave period T, its wavelength L, angular frequency o ¼2p/T, and its direction y. The incident velocity potential of linear waves propagating is given by (Dean and Dalrymple, 2004): igH cosh½kðz þdÞ ikx e fI ¼ 2o coshðkdÞ where g is the gravitational acceleration, d is the water depth, and k is the wave number satisfying the dispersion relation,
o2 ¼ gk tanhðkdÞ
A transparent open sea boundary will not generate any parasitic scattered waves in the computational domain. All the flow approaching these boundaries will be absorbed outside the domain. It is expressed in the following form (Sommerfeld, 1949; outward radiation condition): @f 7 ikf ¼ 0 @x
for
x 81
This radiation condition is applied to open seas and coastlines. The absorbing boundaries are marked in red in Fig. 1. The boundary conditions are presented in the form of Dirichlet type (hf ¼r) at the northern boundary for the incident waves and ! Neuman type ( n ðcrfÞ þ qf ¼ g) along the remaining boundaries for radiation and reflection conditions.
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4. Genetic Algorithms Genetic Algorithms (GA) is a stochastic global search method that operates on a population of potential solutions to produce better approximations to an optimal solution. The population is a set of configurations called chromosomes and the basic GA operators are selection, crossover (interchange of chromosomes segments between mating pairs) and mutation (variation of chromosomes). It produces new individuals that have some parts of both parents genetic material. At each generation, a new set of approximations is created by the process of selecting individuals according to their level of fitness in the problem domain and breeding them together using crossover and mutation operators which are borrowed from natural genetics. This process leads to the evolution of populations of individuals that are better suited to their environment than the individuals that they were created from. GA break the limits of the standard deterministic methods in many optimization problems and is applied: when the search space involves both discrete and continuous domains; when the objective function or the constraints lack regularity; or when the objective function admits a huge number of local optima. Therefore, GA is not only able to improve the solution close to a local optimum, but also to explore a larger extension of the design space for locating the globally optimal solution. Fig. 4 shows the flow chart for the Genetic Algorithm developed for this optimization problem. The developed procedure includes the use of finite element tool for the hydrodynamic problem.
4.1. Fitness function The fitness or cost function F of each individual j is formulated from the objective function and the wave disturbance constraint. The relationship is established as such to penalize the individuals
Fig. 4. Flow Chart for the developed GA.
with high constraint values. F j ¼ f j ðxi Þ þ K½maxð0,C 1 ðxi Þj Þ2 The coefficient K is a factor assigned through trials. 4.2. Search space A search space is defined where all the potential solutions (breakwaters) are generated inside this domain. The domain is defined as convex to ensure that any combination of these generated solutions will remain inside this domain. x1 A S
and
x2 A S,
then
lx1 þ ð1lÞx2 A S
where S represents the convex solution domain. The search space is defined considering the geometrical constraints imposed by the navigational constraint. Fig. 5 shows the domain solution defined considering the constraints for the navigational access channels to sail into the port. By defining this domain solution the navigational constraint has been eliminated from the optimization problem which minimizes the number of constraints in the optimization problem. 4.3. Random generation function The initial population is generated using a random function developed such that all generated individuals in each population can contribute to the convergence of the GA and arbitrarily generated. This random generation function will assist the GA to search in a defined space and enrich the next generation by new feasible solutions. Figs. 5 and 6 shows the random generation inside the design domain for one segment and two segments breakwaters respectively. The efficiency of this approach is illustrated through a comparison with a general algorithm in Section 5. On the basis of the efficiency of each individual, evaluated by the fitness, the developed genetic operator of selection choose the good individuals, that, based on the principle of ‘survival of the fittest’, are destined to the generation of a new population, by using both the genetic operators. Few worse individuals, destined to be modified deeply for the possible random change of all their genes. The next generations have new characteristics that can produce a better solution and however can favor the exploration of the feasible domain, reducing the risk of obtaining only local optima, with respect to traditional algorithms. Particularly the mutation allows to renew the worse individuals destined to extinction, not dispersing their genetic patrimony, and, at the same time, increasing the diversity in the population and thus favoring the exploration of the design domain. The mutation uses
Fig. 5. Domain definition and generation of one segment breakwater.
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the same random generation function applied for the initial population. The operator selects for crossover the round (M/4) best evaluated individuals ranged after the elite and the round (M/6) individuals generated by mutation. The employed strategy involves also the transfer of the best two individuals of each population into the next generation without transformations. Since for problems with few individuals, the best individual(s) is usually transferred, it is believed that the higher the individual number, the higher must be the number of the transferred copies, replacing as many ones extracted randomly, in order to increase the possibility to enhance the population quality and to make the analyses faster. Obviously the copy number must not be too high, in order to avoid that the solution tends to get stuck at a local optimum. In the present paper a copy number of approximately 8% of the individual number has been transferred.
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5. Numerical implementation and results The output of the optimization algorithm is the set of variables, xi ¼ [x1,y1,x2,y2] and xi ¼[x1,y1,x2,y2,x3,y3] for the case of one and two segments breakwaters respectively. These variables determine the optimal layout of the detached breakwater and subsequently an optimal solution for port protection. The port is almost sheltered from the western waves as shown in Fig. 3, but it is exposed to waves coming from the north and north-east. Thus, port protection is required against two incident waves’ conditions. Table 1 shows the critical wave conditions characterized by the significant wave height (Hs) and associated peak period (Tp). The optimization process was performed for two cases: port with a simplified bathymetry (fixed water depth) and port with a real bathymetry. A numerical solution for the hydrodynamic model is implemented in MATLABTM using the Finite Element Method. The optimization process is carried out in MATLABTM using the developed Genetic Algorithm defined in Section 4. The optimization problem is characterized by a highly nonlinear problem. This mainly reverts to the hydrodynamic constraint represented by solving the hydrodynamic model. Each evaluation of the cost function and constraints requires a FEM simulation. A new port layout is defined in each iteration due to the new coordinates of the breakwater nodes defined by the generated population at each time iteration. The time cost is almost similar for each iteration of the optimization problem, since the computational domain is fixed whereas the location and length of the detached breakwater varies.
5.1. Optimization convergence Fig. 6. Domain definition and generation of 2 segment breakwaters.
Table 1 Wave conditions. Wave direction
Hs (m)
Tp (s)
N (01) NE (451)
3 2
9 7
In order to highlight the effectiveness of this optimization approach, a comparison is made with a general algorithm without defining a search space. The comparison is made for the simplified bathymetry case with northern waves. Figs. 7 and 8 show the cost function convergence for the optimization process for the one segment and two segments breakwaters respectively. Like in other optimization algorithms the process is halted when the fitness stops to improve. The algorithm is stopped when a fixed fitness is achieved for 25 iterations for the one segment breakwater case and 35 iterations for the two segment breakwaters.
Fig. 7. Cost function value for the best generation in each population using GA algorithm with general and defined search space (one segment breakwater with northern waves).
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Fig. 8. Cost function value for the best generation in each population using GA algorithm with general and defined search space (two segment breakwaters with northern waves).
Fig. 9. N waves–iteration 2 (one segment breakwater with constant water depth).
Fig. 10. N waves–optimal solution–iteration 10 (one segment breakwater with constant water depth).
5.2. Case 1: simplified bathymetry The comparison curves show significant improvement in the cost function when a defined search space with random generation is introduced in the algorithm. The cost function decreases by 35% and 18% for the one segment and two segment breakwaters respectively. The curves show that several local minima have been visited by the algorithm. For the general algorithm, an optimal solution could be found earlier and it will be difficult to be improved. The defined search space enriches each population with new feasible individuals enabling the algorithm to proceed and not to be captured by the local minima. The more flexible shape of the structure, i.e. increasing the number of variables, makes the optimization problem more complicated. The two segment breakwater has a bigger number of local minima occurring before an optimal solution is found. This shows the difficulty of achieving optimal solutions without using defined search space. The wave height in the computation domain for the optimal solution is shown in Figs. 10 and 14.
5.2.1. Northern waves and one segment breakwater The optimal layout has been determined after ten iterations (Fig. 11). Figs. 9 and 10 show initial and the optimal solution achieved. The shapes developed by the GA are considered reasonable as the random generation started with an inclined shape outside the port and developed to a more horizontal breakwater capable of sheltering the desired area from northern waves. In addition, the optimal breakwater solution is further placed inside the port to reduce wave diffraction effects.
5.2.2. North-eastern waves with one segment breakwater The NE waves are more critical for the port protection and the convergence is slower than N waves. Figs. 11 and 12 present the initial and optimum solution. Various layouts were tested by the
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GA i.e., inclined toward the east, west and north. The optimal layout is similar to the N waves but with a longer breakwater to avoid transmission of NE waves inside the port. 5.2.3. Northern waves with two segments breakwater Other feasible solutions to protect the ports could be two segment breakwaters. This allows the GA to consider different shapes for assessment. Furthermore, this could reduce the wave disturbance as two segment breakwaters has a more flexible shape and will be more efficient to protect the port. Figs. 13 and 14 show selected iterations for the developed layout. It can be recognized that various shapes and lengths have been considered in the different populations. The optimal layout has the minimum length and most acceptable wave disturbance.
Fig. 11. NE waves–iteration 2 (one segment breakwater with constant water depth).
Fig. 12. NE waves–optimal solution–iteration 20 (one segment breakwater with constant water depth).
Fig. 13. N waves–iteration 2 (two segment breakwaters with constant water depth).
5.2.4. North-eastern waves with two segment breakwaters The optimization process gave optimal layout for NE waves in a V shape similar to that for N waves but with longer segments. The robustness of the developed GA can be noticed in the results of the optimal layouts. The two optimal layouts can be compared
Fig. 14. N waves–optimal solution–iteration 24 (two segment breakwaters with constant water depth).
Fig. 15. NE waves–iteration 1 (two segment breakwaters with constant water depth).
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Fig. 16. NE waves–optimal solution–iteration 25 (two segment breakwaters with constant water depth).
Fig. 18. NE waves–optimal solution–iteration 17 (one segment breakwater with real bathymetry).
Fig. 19. NE waves–iteration 1 (two segment breakwaters with real bathymetry). Fig. 17. NE waves–iteration 1 (one segment breakwater with real bathymetry).
One segment breakwater: in Figs. 14–16. The same has been achieved for the optimal layout of the NE waves compared to the N waves (Fig. 17). 5.3. Case 2: port with real bathymetry In order to make this optimization process more practical for real engineering applications, the same process is conducted for the port with the existing bathymetry defined in Fig. 3. The refraction of waves due to changes in water depth inside the port will be introduced due to the bathymetry inclusion. In such problem, the length of the breakwater alone is not sufficient to be optimized but its relative location in the deep water shall be considered as well. A minimum length at a relatively less water depth will be an optimal solution to be determined. Thus the cost function defined previously as minimum length in Section 2 will be modified as minimum area (vertical cross section) to include the water depth where the breakwater will be installed. Subsequently, the optimal placement will minimize the construction costs as well.
f ob ðxi Þ ¼ Minfl1 minðd1 ,d2 Þ þ l1 9d1 d2 9=2g where l1 is the length of the one segment breakwater defined as l1 ¼[(x2 x1)2 þ(y2 y1)2]1/2, d1 and d2 are the corresponding water depth of the nodes 1 and 2 of the segment breakwater respectively. Two segments breakwater: f ob xi Þ ¼ Min
(
) l1 minðd1 ,d2 Þ þ l1 d1 d2 =2 þ l2 minðd2 ,d3 Þ þ l2 d2 d3 =2
where l2 is the length of the second segment breakwater and defined as l2 ¼[(x3 x2)2 þ(y3 y2)2]1/2 and d3 is the corresponding water depth of the node 3 of the second segment breakwater respectively. Based on the optimization results for the simplified bathymetry case, the NE waves have been only selected for the optimization process for the port with real bathymetry case. The one and two segments breakwater have been considered in this case.
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Fig. 20. NE waves–optimal solution–iteration 24 (two segment breakwaters with real bathymetry).
5.3.1. North eastern waves for one and two segment breakwater The wave disturbance inside the port is complicated by introducing the wave refraction along the varying bathymetry. The optimal layout has been obtained after 17 iterations with a one segment breakwater oriented toward the west to avoid penetration of NE waves into the port. The optimal solution has been achieved after 25 iterations for the two segment breakwaters. Figs. 18–20 shows the wave height inside the port for the optimal solutions. The algorithm developed herein provides optimal solution for different configurations of the breakwater and including simplified and real bathymetry.
6. Conclusion An optimization approach has been developed for the design of ports. The aim was to find an optimal layout of a detached breakwater to protect the port subject to wave disturbance and navigational constraints. The breakwater was described by nodes connected by straight edges inside the computational domain whereas the variables are the coordinates of these nodes. The waves’ propagation from deep water has been modeled using the mild slope equation (Berkoff). An optimization process has been introduced using GA integrated with developed generic operators (random generation function, mutation and crossover) suitable for the breakwaters layout. The problem formulation included practical engineering context using real bathymetry application. The results demonstrated the efficiency of an optimization approach that can assess different shapes, location and lengths to determine the optimal layout. The results are satisfactory from the point of view of layout of the detached breakwaters and wave protection.
Further improvements of the hydrodynamic model are suggested to enhance the optimization approach for design of breakwaters in ports. The boussinesq wave model that covers most wave effects of interest in ports, i.e. irregularity of waves, nonlinearity, frequency dispersion and wave–wave interaction is the next step to be incorporated in such approach. In addition, the characterization of the detached breakwater can be improved by increasing the number of nodes describing the structure itself. A larger number of nodes defining the structure will improve the wave attenuation by forming more adequate geometrical shapes. The elaboration of this numerical tool can serve as a conceptual design for the detached breakwaters based on hydrodynamic modelling and optimization algorithms. References Akagi, S., Ito, K., 1984. Optimal design of semisubmersible form by minimizing its motion in random seas. Trans. ASME, J. Mech. Transm. Autom. Des. 106, 23–30. Berkhoff J.C.W 1972. Computation of combined refraction-diffraction. In ASCE(Ed.), Proc.13th Coastal Eng. Conf. , Vancouver, pp. 471–490. Clauss, G.F., Birk, L., 1996. Hydrodynamic shape optimization of large offshore structures. Appl. Ocean Res. 18, 157–171. Dean, R., Dalrymple, R., 2004. Coastal Processes with Engineering Applications. Cambridge University Press, United Kingdom. Elchahal, G., Lafon, P., Younes, R., 2009. Design optimization of floating breakwaters with an interdisciplinary fluid–solid structural problem. Can. J. Civ. Eng. 36, 1732–1743. Isebe, P.A, Bouchette, F., Ivorra, B., Mohammadi, B., 2008a. Shape optimization of geotextiles tubes for sandy beach protection. Int. J. Numer. Methods Eng. 74, 1262–1277. Isebe, D., Azerad, P., Mohammadi, B., Bouchette, F., 2008b. Optimal shape design of defense structures for minimizing short wave impact. Coastal Eng. 55, 35–46. Kagemoto, H., 1992. Minimization of wave forces on an array of floating bodies. Appl. Ocean Res. 14, 83–92. Sommerfeld, A., 1949. Partial Differential Equations in Physics. Academic Press, New York.
Contents lists available at SciVerse ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Optimization of coastal structures: Application on detached breakwaters in ports Ghassan Elchahal a,n, Rafic Younes b, Pascal Lafon c a
Institut Charles Delaunay, University of Technology of Troyes, Troyes, France Lebanese University, Rafic Harriri Campus, Beirut, Lebanon c Institut Charles Delaunay, University of Technology of Troyes, Troyes, France b
a r t i c l e i n f o
abstract
Article history: Received 20 December 2011 Accepted 12 January 2013 Available online 5 March 2013
An optimization approach using Genetic Algorithms (GA) has been developed to determine the optimal layout of detached breakwaters inports subject to wave disturbance and navigational constraints. It aims to modify the shape and location of detached breakwater to achieve an optimal solution. This approach constitutes an advanced step in the design of coastal structures compared to conventional methods. The detached breakwater is parameterized by the coordinates of its nodes which constitute the variables of the optimization problem. The initial population is randomly generated inside a defined search space to ensure large consideration of various potential solutions. Mutation with random generation has been included at each iteration to renew the worse individuals and enrich the GA with new solutions. One and two segment breakwaters have been considered and compared. This optimization approach has been performed for the Port of Beirut as a real engineering application for two cases: simplified and real bathymetry. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Layout of detached breakwaters Water wave mechanics Hydrodynamic modeling Shape optimization Genetic algorithms
1. Introduction The layout of detached breakwaters in ports, i.e. shape and location, is usually designed by assessing several concepts based on the designer engineering experience. The design has not been addressed as a hydrodynamic analysis problem where a solution can be obtained. This may result from several severe problems related to the difficulty of the hydrodynamic analysis and evaluation of such marine structures mainly due to their arbitrarily shapes. It is noticed that optimization applications related to hydrodynamic aspects of marine and coastal structures are rarely reported. However some research work on this subject has been carried out. For example, Akagi and Ito (1984) optimized the heave motion of a hydrodynamic transparent semi submersible using a quadratic programming technique, Kagemoto (1992) optimized the arrangement of vertical floating cylinders in waves, Clauss and Birk (1996) focused on hydrodynamic shape optimization for large offshore structures (oil platforms) based on nonlinear programming algorithms. Elchahal et al. (2009) addressed an optimization problem for floating breakwaters. Isebe et al. (2008a) considered shape optimization approach to tackle beach erosion problems. They addressed the shape optimization of n
Corresponding author. Tel.:þ 33 3251 5651; fax: þ 33 3257 15675. E-mail addresses: [email protected] (G. Elchahal), [email protected] (R. Younes), [email protected] (P. Lafon). 0029-8018/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2013.01.021
geotextile tubes placed on the seabed in order to change the sea bottom shape and subsequently reduce beach erosion effects. Isebe et al. (2008b) also worked on the optimization of defense structures to minimize free surface elevation for fixed bathymetry coastal areas. They determined the optimum dimensions of groynes that yield to minimum wave height. They have used a simplified hydrodynamic model by considering a fixed bathymetry and thus excluded the effects of wave refraction due to water depth variation. To our knowledge this is the first paper to consider an optimization approach for design of breakwaters inside ports with a real engineering application. Detached breakwater in ports are usually designed to attenuate waves from secondary (nondominant) wave directions or for port extension projects. The detached breakwater have numerous advantages, i.e. moving the breakwater seawards allows more space for accommodating splash due to wave overtopping behind the breakwater and this significantly lowers the effects of overtopping at the limit of the working area within the harbor. This can also mean that the crest level of the structure can be reduced. A detached breakwater is likely to be shorter than a shoreconnected one; also the volume of the material needed will be less because of the reduced length and hence significant cost savings can be made. But on the other side the cost of the construction maybe higher because waterborne equipment has to be used. The main objective of this paper is to develop an optimization approach for the design of detached breakwater in ports using
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Genetic Algorithms (GA), i.e. find the optimum layout of the detached breakwater considering wave disturbance inside the port and navigational constraints. The idea reverts to the fact that in such fluid–structure interaction problems where the flow is affected by the presence of a structure in the studied domain, then the structure shape and location can be modified to attain a targeted flow. The optimization approach is defined and elaborated in Section 2. It consists of an objective or cost function, defined by the minimum length and most feasible location of the detached breakwater that could satisfy the wave disturbance criteria and navigational constraints. Wave disturbance inside the port requires a numerical model for wave propagation inside the port. The hydrodynamic model for wave disturbance applied in the optimization problem is defined in Section 3. The Genetic Algorithm is presented in Section 4. Section 5 includes the optimization results for an application for Port of Beirut as a real application, considering two cases for one segment and two segment breakwaters. In addition, a constant water depth bathymetry and the actual bathymetry were considered for each case as well. Finally, recommendations for further developments of optimization algorithm in the marine applications are provided.
Fig. 1. Parameterization of the structure and domain definition. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
2. Optimization problem An optimization problem consists in the minimization of a function fAR, also called cost or objective function, corresponding to the physical criteria to optimize. This function depends on xi, design parameters defining the detached breakwater layout within the admissible set X. In the application presented here, a given set of parameters xi defines a new shape for the breakwater, and thus a new domain Oðxi Þ for the computation. Then by using the hydrodynamic model presented in Section 3, the layout of the detached breakwater is derived and subsequently the value of the cost function f. In addition the free surface elevation Z in O is computed to determine the wave disturbance at a desired location or area inside the port. 2.1. Parameterization of the structure The selection of the optimization parameters is important for the problem definition and shall account for the imposed constraints. It is assumed that no definite shape is preferred and thus the structure is parameterized using the coordinates of the extremity points of the detached breakwater connected by straight edges, as shown in Fig. 1. These nodes, two or three, are free to move in the computational domain to achieve the optimal breakwater layout. Therefore, the numbers of variables will be four or six accordingly, xi ¼[x1,y1,x2,y2] or, xi ¼ ½x1 ,y1 ,x2 ,y2 ,x3 ,y3 The optimization problem is assumed to be finite dimensional constrained minimization problem, which is symbolically expressed as
find a design variable vector xi; to minimize the objective function f(xi) and subjected to the n constraints Cn(xi) r0.
Fig. 2. Port of Beirut–Lebanon.
waves by a long breakwater whereas the Northern Quay is exposed to North and North-Eastern waves and shall be protected with a new detached breakwater. The bathymetry of the Port of Beirut—Lebanon is presented in Fig. 3. It is observed that the water depth varies from 30 m to 5 m inside the port, where it is around 40 m outside the port.
2.3. Objective function The objective is to determine an optimal layout of the detached breakwater, i.e. minimum length and optimum location to achieve an acceptable wave disturbance inside the port. Subsequently an optimum solution will minimize the breakwater costs. The objective function is related to the length of the breakwater and expressed in terms of the geometrical coordinates of the structure. One segment breakwater: f ob ðxi Þ ¼ Min½ðx2 x1 Þ2 þ ðy2 y1 Þ2 1=2
2.2. Bathymetry definition The Port of Beirut—Lebanon (Fig. 2) is located at the eastern coastline of the Mediterranean. The Port is protected from the
Two segments breakwater: ( ) ½ðx2 x1 Þ2 þðy2 y1 Þ2 1=2 f ob ðxi Þ ¼ Min þ½ðx3 x2 Þ2 þ ðy3 y2 Þ2 1=2
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A numerical wave model is required to study the propagation of waves from deep water toward the area of interest inside the port. The well known mild slope equation, derived by Berkhoff (1972), for linear wave propagation has been applied. It combines the resolution of the three main physical mechanisms producing wave transformation: diffraction, reflection and refraction. In general, the diffraction process is of primary importance at the tips of breakwaters or harbor entrances. As harbors are semiclosed wall surrounded areas, the reflection mechanism must also be taken into account. Furthermore, the refraction effect plays a secondary role at areas with a relatively flat bottom, but is very relevant when passing over dredged navigation channels. The mild slope equation is expressed in its elliptic form as follows:
rðCC g rfÞ þ k2 CC g f ¼ 0
Fig. 3. Bathymetry definition of the port.
where (x1,y1), (x2,y2) and (x3,y3) are the coordinates of the extremity nodes of the detached breakwater segments. 2.4. Wave disturbance constraint The wave disturbance in a port is the most significant criteria to be considered in the determining the layout of the facilities prior to construction. The Northern Quay is the most exposed and satisfying the wave disturbance at this desired location will ensure acceptable conditions at other areas in the port. This constraint is expressed in terms of the maximum surface elevation at a desired area in front of the Quay. The allowable wave height constraint is expressed as follows: C 1 ðxi Þ ¼ 2 maxðZðx,y,tÞÞ o a where x and y represent the coordinates of points in the computational domain, and a is the allowable value height at the desired location. A value of a ¼0.3 m has been considered. 2.5. Navigational constraint The location of the new detached breakwater should not have impacts on the navigation inside the port. The ships shall still have sufficient space to sail in and out. The distance between the extremity of the detached breakwater and the existing breakwater shall be greater than a defined distance Dn (Fig. 5). A value of Dn¼ 100 m has been considered in the optimization approach. This constraint is included in the definition of the domain of optimal solutions detailed in Section 4.
where f is the velocity potential, C and Cg are phase and group velocity, respectively. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ð1 þ ð2kd=sinh 2kdÞÞ tanh kd and C g ¼ C C¼ k 2 Solving Berkoff’s mild slope equation allows obtaining the instantaneous velocity potential inside the fluid domain. Then, the free surface elevation with respect to the mean water level at each point within the harbor is easily computed, by the following relation:
Zðx,y,tÞ ¼
io fðx,y,tÞ g
The Berkoff equation is associated with the following boundary conditions related to reflection and radiation phenomena. 3.1. Reflection condition The breakwaters and the Quay walls represent rigid structures. Hence, the flow is reflected on these structures. The fully reflective boundary condition is really inappropriate since in reality partial reflection at the boundaries within a port invariably occurs. Moreover, reflecting boundaries are not always vertical or are fully reflecting. In order to treat these boundaries in the present model, they are schematized as partially reflective and introduced by using a mixed boundary condition instead of the full reflection condition in the manner proposed by Berkhoff (1976). @f kr 1 ¼ ik f, kr þ1 @n where kr represents the reflection coefficient (kr ¼ 0.3) The reflection condition is applied for all the structures marked in blue lines in Fig. 1. 3.2. Radiation condition
3. Hydrodynamic modeling Fluid is assumed to be ideal, flow is considered as irrotational and thus the linear wave theory can be applied. An incident monochromatic wave is characterized by its height H, its wave period T, its wavelength L, angular frequency o ¼2p/T, and its direction y. The incident velocity potential of linear waves propagating is given by (Dean and Dalrymple, 2004): igH cosh½kðz þdÞ ikx e fI ¼ 2o coshðkdÞ where g is the gravitational acceleration, d is the water depth, and k is the wave number satisfying the dispersion relation,
o2 ¼ gk tanhðkdÞ
A transparent open sea boundary will not generate any parasitic scattered waves in the computational domain. All the flow approaching these boundaries will be absorbed outside the domain. It is expressed in the following form (Sommerfeld, 1949; outward radiation condition): @f 7 ikf ¼ 0 @x
for
x 81
This radiation condition is applied to open seas and coastlines. The absorbing boundaries are marked in red in Fig. 1. The boundary conditions are presented in the form of Dirichlet type (hf ¼r) at the northern boundary for the incident waves and ! Neuman type ( n ðcrfÞ þ qf ¼ g) along the remaining boundaries for radiation and reflection conditions.
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4. Genetic Algorithms Genetic Algorithms (GA) is a stochastic global search method that operates on a population of potential solutions to produce better approximations to an optimal solution. The population is a set of configurations called chromosomes and the basic GA operators are selection, crossover (interchange of chromosomes segments between mating pairs) and mutation (variation of chromosomes). It produces new individuals that have some parts of both parents genetic material. At each generation, a new set of approximations is created by the process of selecting individuals according to their level of fitness in the problem domain and breeding them together using crossover and mutation operators which are borrowed from natural genetics. This process leads to the evolution of populations of individuals that are better suited to their environment than the individuals that they were created from. GA break the limits of the standard deterministic methods in many optimization problems and is applied: when the search space involves both discrete and continuous domains; when the objective function or the constraints lack regularity; or when the objective function admits a huge number of local optima. Therefore, GA is not only able to improve the solution close to a local optimum, but also to explore a larger extension of the design space for locating the globally optimal solution. Fig. 4 shows the flow chart for the Genetic Algorithm developed for this optimization problem. The developed procedure includes the use of finite element tool for the hydrodynamic problem.
4.1. Fitness function The fitness or cost function F of each individual j is formulated from the objective function and the wave disturbance constraint. The relationship is established as such to penalize the individuals
Fig. 4. Flow Chart for the developed GA.
with high constraint values. F j ¼ f j ðxi Þ þ K½maxð0,C 1 ðxi Þj Þ2 The coefficient K is a factor assigned through trials. 4.2. Search space A search space is defined where all the potential solutions (breakwaters) are generated inside this domain. The domain is defined as convex to ensure that any combination of these generated solutions will remain inside this domain. x1 A S
and
x2 A S,
then
lx1 þ ð1lÞx2 A S
where S represents the convex solution domain. The search space is defined considering the geometrical constraints imposed by the navigational constraint. Fig. 5 shows the domain solution defined considering the constraints for the navigational access channels to sail into the port. By defining this domain solution the navigational constraint has been eliminated from the optimization problem which minimizes the number of constraints in the optimization problem. 4.3. Random generation function The initial population is generated using a random function developed such that all generated individuals in each population can contribute to the convergence of the GA and arbitrarily generated. This random generation function will assist the GA to search in a defined space and enrich the next generation by new feasible solutions. Figs. 5 and 6 shows the random generation inside the design domain for one segment and two segments breakwaters respectively. The efficiency of this approach is illustrated through a comparison with a general algorithm in Section 5. On the basis of the efficiency of each individual, evaluated by the fitness, the developed genetic operator of selection choose the good individuals, that, based on the principle of ‘survival of the fittest’, are destined to the generation of a new population, by using both the genetic operators. Few worse individuals, destined to be modified deeply for the possible random change of all their genes. The next generations have new characteristics that can produce a better solution and however can favor the exploration of the feasible domain, reducing the risk of obtaining only local optima, with respect to traditional algorithms. Particularly the mutation allows to renew the worse individuals destined to extinction, not dispersing their genetic patrimony, and, at the same time, increasing the diversity in the population and thus favoring the exploration of the design domain. The mutation uses
Fig. 5. Domain definition and generation of one segment breakwater.
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the same random generation function applied for the initial population. The operator selects for crossover the round (M/4) best evaluated individuals ranged after the elite and the round (M/6) individuals generated by mutation. The employed strategy involves also the transfer of the best two individuals of each population into the next generation without transformations. Since for problems with few individuals, the best individual(s) is usually transferred, it is believed that the higher the individual number, the higher must be the number of the transferred copies, replacing as many ones extracted randomly, in order to increase the possibility to enhance the population quality and to make the analyses faster. Obviously the copy number must not be too high, in order to avoid that the solution tends to get stuck at a local optimum. In the present paper a copy number of approximately 8% of the individual number has been transferred.
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5. Numerical implementation and results The output of the optimization algorithm is the set of variables, xi ¼ [x1,y1,x2,y2] and xi ¼[x1,y1,x2,y2,x3,y3] for the case of one and two segments breakwaters respectively. These variables determine the optimal layout of the detached breakwater and subsequently an optimal solution for port protection. The port is almost sheltered from the western waves as shown in Fig. 3, but it is exposed to waves coming from the north and north-east. Thus, port protection is required against two incident waves’ conditions. Table 1 shows the critical wave conditions characterized by the significant wave height (Hs) and associated peak period (Tp). The optimization process was performed for two cases: port with a simplified bathymetry (fixed water depth) and port with a real bathymetry. A numerical solution for the hydrodynamic model is implemented in MATLABTM using the Finite Element Method. The optimization process is carried out in MATLABTM using the developed Genetic Algorithm defined in Section 4. The optimization problem is characterized by a highly nonlinear problem. This mainly reverts to the hydrodynamic constraint represented by solving the hydrodynamic model. Each evaluation of the cost function and constraints requires a FEM simulation. A new port layout is defined in each iteration due to the new coordinates of the breakwater nodes defined by the generated population at each time iteration. The time cost is almost similar for each iteration of the optimization problem, since the computational domain is fixed whereas the location and length of the detached breakwater varies.
5.1. Optimization convergence Fig. 6. Domain definition and generation of 2 segment breakwaters.
Table 1 Wave conditions. Wave direction
Hs (m)
Tp (s)
N (01) NE (451)
3 2
9 7
In order to highlight the effectiveness of this optimization approach, a comparison is made with a general algorithm without defining a search space. The comparison is made for the simplified bathymetry case with northern waves. Figs. 7 and 8 show the cost function convergence for the optimization process for the one segment and two segments breakwaters respectively. Like in other optimization algorithms the process is halted when the fitness stops to improve. The algorithm is stopped when a fixed fitness is achieved for 25 iterations for the one segment breakwater case and 35 iterations for the two segment breakwaters.
Fig. 7. Cost function value for the best generation in each population using GA algorithm with general and defined search space (one segment breakwater with northern waves).
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Fig. 8. Cost function value for the best generation in each population using GA algorithm with general and defined search space (two segment breakwaters with northern waves).
Fig. 9. N waves–iteration 2 (one segment breakwater with constant water depth).
Fig. 10. N waves–optimal solution–iteration 10 (one segment breakwater with constant water depth).
5.2. Case 1: simplified bathymetry The comparison curves show significant improvement in the cost function when a defined search space with random generation is introduced in the algorithm. The cost function decreases by 35% and 18% for the one segment and two segment breakwaters respectively. The curves show that several local minima have been visited by the algorithm. For the general algorithm, an optimal solution could be found earlier and it will be difficult to be improved. The defined search space enriches each population with new feasible individuals enabling the algorithm to proceed and not to be captured by the local minima. The more flexible shape of the structure, i.e. increasing the number of variables, makes the optimization problem more complicated. The two segment breakwater has a bigger number of local minima occurring before an optimal solution is found. This shows the difficulty of achieving optimal solutions without using defined search space. The wave height in the computation domain for the optimal solution is shown in Figs. 10 and 14.
5.2.1. Northern waves and one segment breakwater The optimal layout has been determined after ten iterations (Fig. 11). Figs. 9 and 10 show initial and the optimal solution achieved. The shapes developed by the GA are considered reasonable as the random generation started with an inclined shape outside the port and developed to a more horizontal breakwater capable of sheltering the desired area from northern waves. In addition, the optimal breakwater solution is further placed inside the port to reduce wave diffraction effects.
5.2.2. North-eastern waves with one segment breakwater The NE waves are more critical for the port protection and the convergence is slower than N waves. Figs. 11 and 12 present the initial and optimum solution. Various layouts were tested by the
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GA i.e., inclined toward the east, west and north. The optimal layout is similar to the N waves but with a longer breakwater to avoid transmission of NE waves inside the port. 5.2.3. Northern waves with two segments breakwater Other feasible solutions to protect the ports could be two segment breakwaters. This allows the GA to consider different shapes for assessment. Furthermore, this could reduce the wave disturbance as two segment breakwaters has a more flexible shape and will be more efficient to protect the port. Figs. 13 and 14 show selected iterations for the developed layout. It can be recognized that various shapes and lengths have been considered in the different populations. The optimal layout has the minimum length and most acceptable wave disturbance.
Fig. 11. NE waves–iteration 2 (one segment breakwater with constant water depth).
Fig. 12. NE waves–optimal solution–iteration 20 (one segment breakwater with constant water depth).
Fig. 13. N waves–iteration 2 (two segment breakwaters with constant water depth).
5.2.4. North-eastern waves with two segment breakwaters The optimization process gave optimal layout for NE waves in a V shape similar to that for N waves but with longer segments. The robustness of the developed GA can be noticed in the results of the optimal layouts. The two optimal layouts can be compared
Fig. 14. N waves–optimal solution–iteration 24 (two segment breakwaters with constant water depth).
Fig. 15. NE waves–iteration 1 (two segment breakwaters with constant water depth).
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Fig. 16. NE waves–optimal solution–iteration 25 (two segment breakwaters with constant water depth).
Fig. 18. NE waves–optimal solution–iteration 17 (one segment breakwater with real bathymetry).
Fig. 19. NE waves–iteration 1 (two segment breakwaters with real bathymetry). Fig. 17. NE waves–iteration 1 (one segment breakwater with real bathymetry).
One segment breakwater: in Figs. 14–16. The same has been achieved for the optimal layout of the NE waves compared to the N waves (Fig. 17). 5.3. Case 2: port with real bathymetry In order to make this optimization process more practical for real engineering applications, the same process is conducted for the port with the existing bathymetry defined in Fig. 3. The refraction of waves due to changes in water depth inside the port will be introduced due to the bathymetry inclusion. In such problem, the length of the breakwater alone is not sufficient to be optimized but its relative location in the deep water shall be considered as well. A minimum length at a relatively less water depth will be an optimal solution to be determined. Thus the cost function defined previously as minimum length in Section 2 will be modified as minimum area (vertical cross section) to include the water depth where the breakwater will be installed. Subsequently, the optimal placement will minimize the construction costs as well.
f ob ðxi Þ ¼ Minfl1 minðd1 ,d2 Þ þ l1 9d1 d2 9=2g where l1 is the length of the one segment breakwater defined as l1 ¼[(x2 x1)2 þ(y2 y1)2]1/2, d1 and d2 are the corresponding water depth of the nodes 1 and 2 of the segment breakwater respectively. Two segments breakwater: f ob xi Þ ¼ Min
(
) l1 minðd1 ,d2 Þ þ l1 d1 d2 =2 þ l2 minðd2 ,d3 Þ þ l2 d2 d3 =2
where l2 is the length of the second segment breakwater and defined as l2 ¼[(x3 x2)2 þ(y3 y2)2]1/2 and d3 is the corresponding water depth of the node 3 of the second segment breakwater respectively. Based on the optimization results for the simplified bathymetry case, the NE waves have been only selected for the optimization process for the port with real bathymetry case. The one and two segments breakwater have been considered in this case.
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Fig. 20. NE waves–optimal solution–iteration 24 (two segment breakwaters with real bathymetry).
5.3.1. North eastern waves for one and two segment breakwater The wave disturbance inside the port is complicated by introducing the wave refraction along the varying bathymetry. The optimal layout has been obtained after 17 iterations with a one segment breakwater oriented toward the west to avoid penetration of NE waves into the port. The optimal solution has been achieved after 25 iterations for the two segment breakwaters. Figs. 18–20 shows the wave height inside the port for the optimal solutions. The algorithm developed herein provides optimal solution for different configurations of the breakwater and including simplified and real bathymetry.
6. Conclusion An optimization approach has been developed for the design of ports. The aim was to find an optimal layout of a detached breakwater to protect the port subject to wave disturbance and navigational constraints. The breakwater was described by nodes connected by straight edges inside the computational domain whereas the variables are the coordinates of these nodes. The waves’ propagation from deep water has been modeled using the mild slope equation (Berkoff). An optimization process has been introduced using GA integrated with developed generic operators (random generation function, mutation and crossover) suitable for the breakwaters layout. The problem formulation included practical engineering context using real bathymetry application. The results demonstrated the efficiency of an optimization approach that can assess different shapes, location and lengths to determine the optimal layout. The results are satisfactory from the point of view of layout of the detached breakwaters and wave protection.
Further improvements of the hydrodynamic model are suggested to enhance the optimization approach for design of breakwaters in ports. The boussinesq wave model that covers most wave effects of interest in ports, i.e. irregularity of waves, nonlinearity, frequency dispersion and wave–wave interaction is the next step to be incorporated in such approach. In addition, the characterization of the detached breakwater can be improved by increasing the number of nodes describing the structure itself. A larger number of nodes defining the structure will improve the wave attenuation by forming more adequate geometrical shapes. The elaboration of this numerical tool can serve as a conceptual design for the detached breakwaters based on hydrodynamic modelling and optimization algorithms. References Akagi, S., Ito, K., 1984. Optimal design of semisubmersible form by minimizing its motion in random seas. Trans. ASME, J. Mech. Transm. Autom. Des. 106, 23–30. Berkhoff J.C.W 1972. Computation of combined refraction-diffraction. In ASCE(Ed.), Proc.13th Coastal Eng. Conf. , Vancouver, pp. 471–490. Clauss, G.F., Birk, L., 1996. Hydrodynamic shape optimization of large offshore structures. Appl. Ocean Res. 18, 157–171. Dean, R., Dalrymple, R., 2004. Coastal Processes with Engineering Applications. Cambridge University Press, United Kingdom. Elchahal, G., Lafon, P., Younes, R., 2009. Design optimization of floating breakwaters with an interdisciplinary fluid–solid structural problem. Can. J. Civ. Eng. 36, 1732–1743. Isebe, P.A, Bouchette, F., Ivorra, B., Mohammadi, B., 2008a. Shape optimization of geotextiles tubes for sandy beach protection. Int. J. Numer. Methods Eng. 74, 1262–1277. Isebe, D., Azerad, P., Mohammadi, B., Bouchette, F., 2008b. Optimal shape design of defense structures for minimizing short wave impact. Coastal Eng. 55, 35–46. Kagemoto, H., 1992. Minimization of wave forces on an array of floating bodies. Appl. Ocean Res. 14, 83–92. Sommerfeld, A., 1949. Partial Differential Equations in Physics. Academic Press, New York.