Narrow Results

Topology optimization for fluid flow aims at finding the location of a porous medium minimizing a cost functional under constraints given by the Navier–Stokes equations. The location of the porous media is usually taken into account by adding a penalization term $\alpha u$, where $\alpha$ is a kinematic viscosity divided by a permeability and $u$ is the velocity of the fluid. The fluid part is obtained when $\alpha =0$ while the porous (solid) part is defined for large enough $\alpha$ since this formally yields $u=0$. The main drawback of this method is that only solid that does not let the fluid to enter, that is perfect solid, can be considered. In this paper, we propose to use the porosity of the media as optimization parameter hence to minimize some cost function by finding the location of a porous media. The latter is taken into account through a singular perturbation of the Navier–Stokes equations for which we prove that its weak-limit corresponds to an interface fluid-porous medium problem modeled by the Navier–Stokes–Darcy equations. This model is then used as constraint for a topology optimization problem. We give necessary condition for such problem to have at least an optimal solution and derive first order necessary optimality condition. This paper ends with some numerical simulations, for Stokes flow, to show the interest of this approach.

We discuss the inverse problem associated with the identification of the location and shape of a scatterer fully embedded in a homogeneous halfplane, using scant surficial measurements of its response to probing scalar waves. The typical applications arise in soils under shear (SH) waves (antiplane motion), or in acoustic fluids under pressure waves. Accordingly, we use measurements of either the Dirichlet-type (displacements), or of the Neumann-type (fluid velocities), to steer the localization and detection processes, targeting rigid and sound-hard objects, respectively. The computational approach for localizing single targets is based on partial-differential-equation-constrained optimization ideas, extending our recent work from the full-¹ to the half-plane case. To improve on the ability of the optimizer to converge to the true shape and location we employ an amplitude-based misfit functional, and embed the inversion process within a frequency- and directionality-continuation scheme, which seem to alleviate solution multiplicity. We use the apparatus of total differentiation to resolve the target's evolving shape during inversion iterations over the shape parameters, à la.^2,3 We report numerical results betraying algorithmic robustness for both the SH and acoustic cases, and for a variety of targets, ranging from circular and elliptical, to potato-, and kite-shaped scatterers.