Minimal transport networks with general boundary conditions

Vascular networks are used across the kingdoms of life to transport fluids, nutrients and cellular material. A popular unifying idea for understanding the diversity and constraints of these networks is that the conduits making up the network are organized to optimize dissipation or other functions within the network. However the general principles governing the optimal networks remain unknown. In particular Durand showed that under Neumann boundary conditions networks, that minimize dissipation should be trees. Yet many real transport networks, including capillary beds, are not simply connected. Previously multiconnectedness in a network has been assumed to provide evidence that the network is not simply minimizing dissipation. Here we show that if the boundary conditions on the flows within the network are enlarged to include physical reasonable Neumann and Dirichlet boundary conditions (i.e. constraints on either pressure or flow) then minimally dissipative networks need not be trees. To get to this result we show that two methods of producing optimal networks, namely enforcing constraints via Lagrange multipliers or via penalty methods, are equivalent for tree networks.