Constructal optimization of nonuniformly distributed tree-shaped flow structures for conduction

Constructal tree designs are hierarchical high-conductivity paths that minimize the global resistance between an entire volume and one point. In past work, the structure was optimized as a sequence of building blocks (volume sizes), which started with the smallest size (elemental volume) and continued toward larger and more complex assemblies (first construct, second construct, etc.). The resulting structure had a `uniform' distribution of interstitial spaces, because the size of the elemental volume was fixed. In this paper we relax the elemental size constraint, and show that the added design freedom leads to significant improvements in global performance, i.e., to decreases in the global resistance to volume–point flow. Each tree structure, or the distribution of high-conductivity material through low-conductivity background, is optimized by simulating numerically and comparing large numbers of designs where the geometry changes smoothly from one design to the next. The results show that each optimized structure has not one but several elemental volume sizes, and that the volume elements situated far from the root of the tree are notably smaller. The resulting tree is nonuniform, i.e., denser near the periphery of its canopy. In sum, better global performance is achieved when the complexity and number of degrees of freedom of the structure are increased. In the same direction, the optimized nonuniform tree structure looks more and more natural.