Table 1.

Comparison between previous works on optimal transport networks and the results presented in this work. Previous works focus on optimizing dissipation either alone or in combinations with damage resistance or flow fluctuations (first two rows). They constrain the network by imposing a fixed material cost of vessels. Our work agrees with previous work on the morphologies of minimally dissipative networks under material constraint (first row), but extends the classes of the target functions and constraints. Specifically we study uniform flow networks, under both material constraint and a Murray constraint that includes dissipation (last four rows).

Target functional	Constraint	Method
$\sum \frac{Q_{kl}^2}{{\kappa }_{kl}}$	$\sum {\kappa }_{kl}^{\gamma }$	local topological optimization [16], global optimization [6], structural adaptation [22], growth and structural adaptation [44], Section 3.1
$\sum \frac{Q_{kl}^2}{{\kappa }_{kl}}$ with damage and flow fluctuations	$\sum {\kappa }_{kl}^{\gamma }$	global optimization [26], fluctuating source [13]
$\sum \frac{1}{2}{Q}_{kl}^2$	$\sum {\kappa }_{kl}^{\gamma }$	Section 3.2
$\sum \frac{1}{2}{(Q_{kl}-\stackrel{‒}{Q})}^2$	$\sum {\kappa }_{kl}^{\gamma }$	Section 3.3
$\sum \frac{1}{2}{Q}_{kl}^2$	$\sum \left({\kappa }_{kl}^{\gamma }+a\frac{Q_{kl}^2}{{\kappa }_{kl}}\right)$	Section 3.4
$\sum \frac{1}{2}{(Q_{kl}-\stackrel{‒}{Q})}^2$	$\sum \left({\kappa }_{kl}^{\gamma }+a\frac{Q_{kl}^2}{{\kappa }_{kl}}\right)$	Section 3.4