In this study, topology optimisation for heat-conducting paths in a three-dimensional domain was investigated. The governing equations for the temperature distribution were solved using the finite volume method, the sensitivities of the objective function (average temperature) were solved using the adjoint method, and finally, the optimal architecture was found with the method of moving asymptotes (MMA) using a self programmed code. A two-dimensional domain was evaluated first as a validation for the code and to compare with other papers before considering a three-dimensional cubic domain.
For a partial Dirichlet boundary, it was found that the converged architecture in three dimensions closely resembled the converged architectures from two dimensions, with the main branches extending to the outer corners of the domain. However, the partial Dirichlet boundary condition was not realistic, and to represent a more realistic case, a full Dirichlet boundary was also considered.
In order to investigate a full Dirichlet boundary condition, the domain had to be supplied with an initial base for the architecture to allow variation in the sensitivities. It was found that the width and height of this base had a significant effect on the maximum temperature. A height of 0.04 with a base width of 0.24 proved to be the most effective, since this small base gave the MMA enough freedom to generate a tree structure. It was first assumed that this base should be in the centre of the bottom boundary and this was later proved. The results showed again that the maximum temperature decreased with an increase in the conductivity ratio or volume constraint. The architectures were similar to the partial Dirichlet boundary, again with the main branches extending to the outer corners of the domain. The main branches were thinner compared with the partial Dirichlet boundary and fewer secondary branches were observed. It was concluded that a full Dirichlet boundary could be solved using topology optimisation, if the boundary was supplied with an initial base.
With the successful implementation of the full Dirichlet boundary with one initial base, multiple bases were investigated. First, two bases were used and it was assumed that the optimal placement for these bases was in the centre of each respective half of the bottom boundary, which was later confirmed. The optimal width and height of 0.24 and 0.04 respectively were again optimal for each specific base. The same procedure was followed for four bases and it was assumed that the optimal placement was in the centre of each respective quadrant of the bottom boundary, which was also later confirmed. The optimal width and height of 0.12 and 0.04 respectively were found for this case. With this established, optimisation runs for different conductivity ratios and volume constraints were completed for two and four bases. It was found that two bases offered increased performance in terms of the maximum temperature compared with one base. An increase in performance was also observed when using four bases compared with two bases. A maximum of 20.4% decrease in the maximum temperature was observed when comparing four bases with one.
Keywords: topology, optimisation, conduction, three-dimensional