Topology optimization of flow networks Original Research Article

The field of topology optimization is well developed for load carrying trusses, but so far not for other similar network problems. The present paper is a first study in the direction of topology optimization of flow networks. A linear network flow model based on Hagen–Poiseuille’s equation is used. Cross-section areas of pipes are design variables and the objective of the optimization is to minimize a measure, which in special cases represents dissipation or pressure drop, subject to a constraint on the available (generalized) volume. A ground structure approach where cross-section areas may approach zero is used, whereby the optimal topology (and size) of the network is found. A substantial set of examples is presented: small examples are used to illustrate difficulties related to non-convexity of the optimization problem; larger arterial tree-type networks, with bio-mechanics interpretations, illustrate basic properties of optimal networks; the effect of volume forces is exemplified. We derive optimality conditions which turns out to contain Murray’s law; thereby, presenting a new derivation of this well known physiological law. Both our numerical algorithm and the derivation of optimality conditions are based on an ε-perturbation where cross-section areas may become small but stay finite. An indication of the correctness of this approach is given by a theorem, the proof of which is presented in an appendix. Article Outline