Topology optimization of flow networks

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 epsilon-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. (C) 2003 Elsevier B.V. All rights reserved.