Shape optimization of geotextile tubes for sandy beach protection Damien Isebe†,∗, Pascal Azerad†, Frederic Bouchette♭, Benjamin Ivorra♮ and Bijan Mohammadi†. † Institut de Mathématiques et de Modélisation de Montpellier, Université de Montpellier II, 34095 Montpellier, France ♭ Montpellier Geosciences, Université de Montpellier II, 34095 Montpellier, France ♮ Departamento de Matemática Aplicada, Facultad de Ciencias Matemáticas, 28040 Madrid, Spain Draft for IJNME Abstract The paper describes how to tackle new challenging coastal engineering problems related to beach erosion with a shape optimization approach. The method modifies the shape of the sea bottom in order to reduce beach erosion effects. Global optimization is shown to be necessary as the related functionals have several local minima. We describe the physical model used, the proposed protection devices against beach erosion and real case applications. Keywords. Shape design, global optimization, water wave propagation, scat- tering, coastal engineering, beach erosion. 1 Introduction Beach erosion problems bring increasing engineering demand. Indeed, about 70% of world beaches are crossing an erosion phase, 20% are stable and 10% show signs of fattening [1, 2]. Obviously, this has major economical and envi- ronmental impacts. Groins, breakwaters and other coastal structures are used to decrease water wave energy or to control sediment flows. The shape of these devices are usually determined using simple hydrodynamical assumptions, struc- tural strength laws or empirical considerations. However, as we will see, these are ∗Email: isebe@math.univ-montp2.fr 1 not fully satisfactory because of secondary effects. Our aim is to take advantage of shape optimization techniques, mainly used in aeronautics [3, 4], to propose new solutions to tackle this problem. This approach is fully innovative in coastal engineering. Also, efficient global optimization algorithms are necessary to avoid the design to converge to local minima. Indeed, we will see that the related functionals have several local minima. Moreover, the search space is often non-connected. We use a semi-deterministic algorithm to allow global optimization of systems governed by PDEs with a low calculation complexity [5, 6]. The paper is structured as follows. In section 2, we recall the state of the art on geotextile tubes and we describe their parameterization. Section 3 presents the flow model used for the water waves propagation. Section 4 is dedicated to the description of the minimization problem. Finally, section 5 displays and discusses optimization results for two beach protection studies in Northwestern Mediterranean sea with the aim of reducing the energy available for sediment transport. In the Appendix, we briefly expose the optimization method used. 2 Geotextile tubes and erosion process Water waves propagating toward the coast are characterized by their height H, their period T and their direction θ. The period does not change during the propagation but the direction and the height may vary when approaching emerged or submerged structures. This is mainly due to scattering phenomenon. Knowing the height H of the wave is crucial for the study of an erosion problem. Indeed, the suspension of sediments, produced by water wave action, is the main mechanism of erosion process. The suspension of sediments is essentially linked to the associated water wave mechanical energy E = 1 8 ρgH2 where ρ is the water density and g the gravity acceleration [7, 8, 9]. Oceanographer observations of erosion show that the water waves can be roughly sorted in two categories according to their height H, below or above a critical value Hlim. Basically, those above Hlim, mainly present during storms, are erosive. They generate large mechanical energy. On the other hand, when H < Hlim, waves foster the reconstruction of eroded beaches. In that follows, the first class of water waves will be called erosive and the second class constructive. In order to decrease water waves impact along the coastline many structures have been proposed [10, 11]. Until recently the most used are emerged break- waters or groins built with rocks or concrete. However, these techniques are expensive and only short-term solution for the beach protection because they mainly transfer in space the erosion process (figure 1 shows the negative impact of emerged groins on erosion. Accretion occurs only upstream from the longshore 2 drift, whereas erosion is amplified downstream. Hence, it is soon necessary to build another groin further downstream... and so on). Currently, interest goes to a new generation of soft structures having less im- pact on coastal hydro-sedimentary system [12, 13, 14]. These devices are geotex- tile tubes, also called geotubes (figure 2-Up). These geotubes are long cylinders made of synthetic textile and filled up with sand. This paper discusses shape and location optimizations of geotubes for two sites both located on the Northwestern Mediterranean French coast. The first analysis concerns the protection of a 2.4 km beach located between Sète and Agde [15] (figure 2-Down-(Left)). This is an large-scale industrial project under strict feasibility constraints. The Bas-Rhône Languedoc Company (BRL) is in charge of the device layout and installation. This company has a great experience of land-use management and development in Languedoc-Roussillon (a french area) in order to perform engineering studies and advisory services. Concerning the studied site, the coastal zone is characterized by a very low tidal excursion and moderate waves. This beach is subjected to severe erosion and the coastline has recorded a shoreward displacement of about 50 m since 1967 with a rate of shore retreat of about 1 m/year since 10 years [16]. In this spot, the critical wave height Hlim is about 2 m. The second situation concerns the protection of the Le Grau du Roi Le Bou- canet beach (figure 2-Down-(Right)). This problem leads to a higher dimension design problem, as we shall see below. 2.1 Geotube parameterization Along the barred beach between Sète and Agde, the coastal topobathymetric profile includes two natural sandbars, created under the water wave action. An engineering preliminary study suggested first to restructure the beach and the two natural sandbars by sand recharging and secondly to place two geotubes side-by- side behind the second natural sandbar in order to protect the new beach [15]. In the sequel, we model the two geotubes as a single one twice larger. Figure 3 shows a cross-section of the initial topobathymetry, the suggested sand recharging and the location of the two geotubes. The initial topobathymetric data available for this study consists of series of echosounding data obtained from numerous monitorings since 2000 [16, 17]. So, we recreate exactly the real morphodynamic of the spot. Geotubes will merely be represented by local modification of the topobathym- etry. More precisely, in each node of the discrete domain we have a given value for the initial topobathymetry. To add a geotube arbitrarily in the domain, we parameterize its position using a series of N control points in the domain. Cubic splines are used to connect these points. Once the location known, the 3 shape (cross-section) of the geotube is given by a Gaussian function of the form f(d) = He−sd2 where d is the normal distance to the spline. This adds two addi- tional parameters (s,H) for the control of the height and width of the geotubes. Thus, we have in each node a modification of the topobathymetry (see figure 4) which accounts for the presence of the geotubes. 3 Flow model The propagation of water waves over linear irregular bottom bathymetry and around islands involves many processes such that shoaling, refraction, energy dissipation and diffraction. To compute the water wave propagation, we use the REF/DIF code [18, 19, 20]. REF/DIF is a open source software designed for modelling wave propaga- tion over a weakly varying seabottom. It takes into account both refraction and diffraction phenomena or wave attenuation. However, this model does not ac- count for the reflection phenomenon. This appears, for instance, in water wave propagation in a harbor with vertical emergent structures. Our applications only concern propagation toward sandy beaches. The model has been validated on various experimental test cases [21, 22, 23]. REF/DIF is based on a parabolic weakly nonlinear combined refraction and diffraction model which incorporates all of the effects mentioned above [24, 20]. This model combines the essential features of the two specific approaches, a mild slope model [25, 26, 27, 28] and a diffraction model [29, 30]. In this section, we briefly present the nonlinear combined refraction/diffraction model. 3.1 Nonlinear combined refraction/diffraction model Kirby & Dalrymple, at the Center for Applied Coastal Research (University of Delaware, US), developed a general formulation governing waves in a domain with slow but arbitrary depth variations [24, 31]. The following parabolic approxima- tion for refraction/diffraction is obtained, 2ikCCg ∂A ∂x + 2k (k − k0) (CCg) A + iA ∂ (kCCg) ∂x (1) + ∂ ( CCg ∂A ∂y ) ∂y − k (CCg) K ′|A|2A = 0 where C = √ g k tanh kh (phase speed), (2) 4 Cg = C (1 + 2kh sinh 2kh ) 2 (group velocity), (3) K ′ = k3 ( C Cg ) cosh 4kh + 8 − 2 tanh2 kh 8 sinh4 kh , (4) h(x, y) is the local water depth and g the gravity. The local wave number, k(x, y), is related to the angular frequency of the waves, ω, and the water depth h by the nonlinear dispersion relationship ω2 = gk tanh kh. (5) k0 is a reference wave number related to the incoming condition. Equation (1) is valid under the mild slope assumption ∇h/kh << 1 [24]. Equations (1) and (5) are numerically solved by an implicit finite difference method [19, 20] in a domain of size [0,mr] × [0, nr]. We define xi = (i − 1) ∗ ∆x and yj = (j − 1) ∗ ∆y where ∆x and ∆y are the space steps on each direc- tions. The axis x = 0 is the offshore boundary where the incoming condition is prescribed. The water wave propagation is computed for the increasing i. The lateral boundaries y = 0 and y = nr have open boundary conditions. Concern- ing the topobathymetry, at each node (i, j) of the grid, we have a positive value h(i, j) corresponding to the water depth. In the model (1), A = A(x, y) is the complex amplitude related to the water surface displacement by η(x, y) = ℜ(A(x, y)eikx). (6) 4 Cost function An optimization problem refers to the minimization of a cost function J , also called objective function (see the Appendix ). In this study, our aim is to optimize the shape of a given geotube and its distance to the coast in order to minimize the energy available for sediment transport in the near-shore zone. We have seen in section 2 that this cost function can be expressed as a function of the water wave mechanical energy E = 1 8 ρgH2 where ρ is the water density and H = 2A [7, 8]. This energy is crucial in the erosion process as it fosters the motion of sediments. Another important issue for the modified geometry (i.e. after addition of the geotube) is that the geotube should not increase the sea bottom fluid particle ve- locity1 shoreward. For a water wave propagation in the x-direction, the stationary 1Also called bottom orbital velocity 5 bottom orbital velocity Uorb is given by [30] Uorb = Agk ω cosk · x (7) with x = (x, y) and k = k(cos θ, sin θ) where k is the wave number and θ the wave direction. However, section 2 and Table 1 suggest two main categories of water waves: the constructive and the erosive ones which, as we said, correspond to waves with heights below and above the critical water height Hlim = 2 meters. Hence, for a given direction of propagation θ, the following cost function is considered Jθ = ∫ D EH>Hlim dS ∫ D EH