  In this paper, we study the role of boundary conditions on the optimal shape
of a dyadic tree in which flows a Newtonian fluid. Our optimization problem
consists in finding the shape of the tree that minimizes the viscous energy
dissipated by the fluid with a constrained volume, under the assumption that
the total flow of the fluid is conserved throughout the structure. These
hypotheses model situations where a fluid is transported from a source towards
a 3D domain into which the transport network also spans. Such situations could
be encountered in organs like for instance the lungs and the vascular networks.
Two fluid regimes are studied: (i) low flow regime (Poiseuille) in trees with
an arbitrary number of generations using a matricial approach and (ii) non
linear flow regime (Navier-Stokes, moderate regime with a Reynolds number 100)
in trees of two generations using shape derivatives in an augmented Lagrangian
algorithm coupled with a 2D/3D finite elements code to solve Navier-Stokes
equations. It relies on the study of a finite dimensional optimization problem
in the case (i) and on a standard shape optimization problem in the case (ii).
We show that the behaviours of both regimes are very similar and that the
optimal shape is highly dependent on the boundary conditions of the fluid
applied at the leaves of the tree.
