An optimal shape problem related to the realistic design of river fishways

A river fishway is a hydraulic structure that facilitates fish in overcoming obstacles (dams, waterfalls, etc.) to their spawning and other migrations in rivers. In this work we present a mathematical formulation of an optimal design problem for a vertical slot fishway, where the state system is given by the 2D shallow water equations fixing the height and velocity of water, the design variables are the geometry of the slots, and the objective function is determined by the existence of rest areas for fish and of a water velocity suitable for fish swimming capability. We also derive an expression for the gradient of the objective function via the adjoint system. From the numerical point of view, we present a characteristic-Galerkin method for solving the shallow water equations, and a direct search algorithm for the computation of the optimal design variables. Finally, we give numerical results obtained for a standard ten pools channel.