On the non-optimality of tree structures for heat conduction

• Published in: International journal of heat and mass transfer 2018-07, Vol.122, p.660-680
• Main Authors: Yan, Suna, Wang, Fengwen, Sigmund, Ole
• Format: Article
• Language: English
• Subjects: Conduction heating; Conductive heat transfer; Genetic algorithms; Heat conduction; Heat conductivity; Heat transfer; Interpolation; Lamellar needle structures; Lamellar structure; Mathematical models; Microstructure; Pipes; Sheet modelling; Topology; Topology optimization; Tree structures; Variable thickness; Volume-to-point problem
• ISSN: 0017-9310; 1879-2189
• DOI: 10.1016/j.ijheatmasstransfer.2018.01.114

•Tree-like structures are not optimal for thermal volume-to-point problems.•Lamellar, needle-like structures are the true optima for volume-to-point problems.•Most topology optimization approaches converge to local tree-like minima.•Non-global optima can be alleviated by use of initial guesses from rank-1 designs.•Rank-1 and finite lamellar designs are better than results from the literature. This paper revisits topology optimization of heat conduction structures for minimum thermal compliance and minimum maximum temperature, respectively. For both optimization problems, volume-to-line and volume-to-point structures are optimized based on three material interpolation models describing different design spaces regarding the relation between material density and effective conductivity. The numerical results are backed up by analytical studies. Comparisons of results show that lamellar needle structures, rather than commonly seen tree structures, constitute the optimal topologies for heat conduction. This contradicts the usual hypothesis drawn from the observation of natural transferring systems and designs from numerous related studies. The conclusion still holds when a minimum length scale is imposed for both high and low conductive phases. Finally, the minimum thermal compliance problem and the min-max temperature problem are compared in terms of optimal microstructures. Lamellar microstructures with the normal to the material layers bisecting the gradients of direct and adjoint solutions are optimal for both types of problems. The variable thickness sheet model is optimal only for the self-adjoint minimum thermal compliance problem.