Structure of Optimal Transport Networks Subject to a Global Constraint

The structure of pipe networks minimizing the total energy dissipation rate is studied analytically. Among all the possible pipe networks that can be built with a given total pipe volume (or pipe lateral surface area), the network which minimizes the dissipation rate is shown to be loopless. Furthermore, such an optimal network is shown to contain at most $N-2$ nodes in addition to the $N$ sources plus sinks that it connects. These results are valid whether the possible locations for the additional nodes are chosen freely or from a set of nodes (such as points of a grid). Applications of these results to various physical situations and to the efficient computation of optimal pipe networks are also discussed.

Figure 1

Shift of material in a loop of the network: (a) the original loop, where flow directions in each pipe are indicated with arrows; (b) the same loop, where the cross-sectional area of a pipe is increased when the direction of its carried flow is $\mathcal{A}\to \mathcal{B}$ along path ($\alpha$) or $\mathcal{B}\to \mathcal{A}$ along path ($\beta$), and decreased otherwise. The other cross-sectional areas in the network remain unaltered.

Figure 2

(a) The two adjoining pipes of a twofold junction carry flows in opposite directions in order to satisfy flow rate conservation. (b) The two adjoining pipes can be favorably replaced with a straight one: since the total pipe length is shortened, the dissipation rate will be decreased for a fixed value of $C_n$.