Automatic Adjoint-based Optimisation of Drag-Parametrised Coastal Protection Layout

Coastal areas are prone to hazards from the sea, for example, coastal erosion, storm surge and sea level rise. To protect coastal settlements, coastal protection strategies e.g. mangrove forest, breakwaters and coastal rock armour units (AUs), are used. With limited time and resources, it is important to consider the optimal layout of these structures so as to maximise the protection.

Numerical modelling is often used to assess the response of designs. One computationally efficient approach is to consider the protection structures as a drag term and establish the linkage by parametrising the drag coefficient with the structures. Still, finding optimal design manually is time-consuming and labour-intensive, as there are too many possible designs.

Such a problem is inherently an optimisation problem, constrained by the dynamical equations, e.g. shallow water equations. In fact, representing the structures as drag allows the optimisation with an adjoint, and the optimal design can be numerically found by gradient-based optimisation. Funke et al. [1] uses this method to determine the optimal turbine density that maximises the profit. Similar approach can be adopted to find the optimal “coastal defence density”. Though adjoint models are tedious to implement, numerical tools like Firedrake [4] and pyadjoint [3] have been developed to facilitate the automatic generation of forward and discrete adjoint model. This capability is also demonstrated by Kärnä et al. [2], in calibrating the spatially-varying drag coefficient of a two-dimensional shallow water model (SWE).

Figure 1: Schematics of the experimental setup. Idealised tide forced at left and right boundaries. u_t ≈ [sin(ωt+ Φ), 0]^T , where ω and Φ are the angular frequency and phase of the tide. Material can only placed within the “Deploy Extent”. ∂Ω: Perimeter of the island.

 

Combining the idea and implementation, we demonstrate this adjoint-based optimisation in an idealised scenario, where a circular island is placed inside a tidal-forced rectangular channel (Fig. 1). We attempt to parametrise the density of AUs (ρ_cdef ) as an additional frictional stress of the SWE. With the objective to minimise the total kinetic energy (KE) around the perimeter of the circular island, while constrained by the dynamics, materials, locations and bounds, optimal layout can be numerically found (Fig. 2). This approach can work with other drag parametrisation, and can be easily extended to larger coastal models for realistic applications, which is the future direction of this research. The optimised results can provide engineers qualitative and quantitative estimates of the placement, and also suggest good starting layouts for further high-fidelity modelling, reducing much of manual effort and computational cost.

Figure 2: Resembling a flood-dominated system. Simulation results during (a) ebb and (b) flood tides, using optimised layout from (c). (d) Objective functional during optimisation. (e) P_KE = ½ ∫_∂Ω  ρ₀ (u · u) ds, with protection (Opt.) and without protection (No def.).

 

 

References

[1] S.W. Funke et. al. (2016). DOI: 10.1016/j.renene.2016.07.039.

[2] Tuomas Kärnä et. al. (2023). DOI: 10.1029/2022MS003169.

[3] Sebastian K. Mitusch et. al. (2019). DOI: 10.21105/joss.01292.

[4] Florian Rathgeber et. al. (2016). DOI: 10.1145/2998441