Topology optimization of flow networks

• Published in: Computer methods in applied mechanics and engineering Vol. 192; no. 35; pp. 3909 - 3932
• Main Authors: Klarbring, Anders, Petersson, Joakim, Torstenfelt, Bo, Karlsson, Matts
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.01.2003; Elsevier
• Subjects: Biological and medical sciences; Biomechanics. Biorheology; Blood vessels and receptors; Exact sciences and technology; Fundamental and applied biological sciences. Psychology; Fundamental areas of phenomenology (including applications); Physics; Solid mechanics; Static elasticity; Static elasticity (thermoelasticity...); Structural and continuum mechanics; TECHNOLOGY; TEKNIKVETENSKAP; Tissues, organs and organisms biophysics; Vertebrates: cardiovascular system; 62; 29; 61; Human; Measurement; Pipe flow; Numerical method; Topology; Modeling; Optimization; Biomechanics; Perturbation method; Truss structure; Duct network; Poiseuille flow; Circulatory system
• ISSN: 0045-7825; 1879-2138; 1879-2138
• DOI: 10.1016/S0045-7825(03)00393-1

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.