Topology optimization of flow networks

✍ Scribed by Anders Klarbring; Joakim Petersson; Bo Torstenfelt; Matts Karlsson

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 788 KB
• Volume: 192
• Category: Article
• ISSN: 0045-7825
• DOI: 10.1016/s0045-7825(03)00393-1

✦ Synopsis

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-PoiseuilleÕs 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 MurrayÕs 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 e-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.