Architecture of optimal transport networks.

We analyze the structure of networks minimizing the global resistance to flow (or dissipative energy) with respect to two different constraints: fixed total channel volume and fixed total channel surface area. First, we show that channels must be straight and have uniform cross-sectional areas in such optimal networks. We then establish a relation between the cross-sectional areas of adjoining channels at each junction. Indeed, this relation is a generalization of Murray's law, originally established in the context of local optimization. We establish a relation too between angles and cross-sectional areas of adjoining channels at each junction, which can be represented as a vectorial force balance equation, where the force weight depends on the channel cross-sectional area. A scaling law between the minimal resistance value and the total volume or surface area value is also derived from the analysis. Furthermore, we show that no more than three or four channels meet at each junction of optimal bidimensional networks, depending on the flow profile (e.g., Poiseuille-like or pluglike) and the considered constraint (fixed volume or surface area). In particular, we show that sources are directly connected to wells, without intermediate junctions, for minimal resistance networks preserving the total channel volume in case of plug flow regime. Finally, all these results are compared with the structure of natural networks.