On projection methods, convergence and robust formulations in topology optimization

Abstract

Mesh convergence and manufacturability of topology optimized designs have previously mainly been assured using density or sensitivity based filtering techniques. The drawback of these techniques has been gray transition regions between solid and void parts, but this problem has recently been alleviated using various projection methods. In this paper we show that simple projection methods do not ensure local mesh-convergence and propose a modified robust topology optimization formulation based on erosion, intermediate and dilation projections that ensures both global and local mesh-convergence.

Appendix

The thresholds for the dilated and the eroded designs can be chosen independently, if different length scales need to be imposed on the void and the solid phases. The procedure for determining the minimum length scale is similar to the one presented in Section 6. The estimation is based on the assumption that the topology does not change between the three designs—the eroded, the intermediate and the dilated. Two cases for the filtered designs are shown in Fig. 19. In the first case, Fig. 19a, the maximum of the filtered design ${\bar{\tilde{\rho}}}$ is equal to the threshold for the eroded design and 0/1 projection with threshold η _e results in point with zero length scale. In the second case, Fig. 19b, the maximum of the filtered design is below η _e , and the projection with threshold η _e results in void phase. Therefore, even if the projections with thresholds η ⁱ and η ^d result in intervals with finite lengths, the topology for ${\bar{\tilde{\rho}}}^e, {\bar{\tilde{\rho}}}^i$ and ${\bar{\tilde{\rho}}}^d$ is different and the presented minimum length scale estimate is not valid for this case.

Fig. 19

Filtered density field and with 0/1 projection based on different thresholds

If the topologies are the same for all three thresholds η ^e, η ⁱ and η ^d, $\tilde{\rho}_{\max}$ is always $\tilde{\rho}_{\max}\geq \eta_e$. Then the minimum length scale for the intermediate and the dilated designs can be estimated numerically by finding the lengths b and b ^d of the intervals obtained by crossing the threshold lines with the filtered density. The length scale for the void phase can be estimated in a similar way. The difference between η ^e and η ⁱ determines the length scale in the solid phase and the difference between η ^d and η ⁱ determines the length scale in the void phase. Graphs for the normalized length scales of the solid and void phases as functions of the thresholds are shown in Figs. 20 and 21.

Fig. 20

Length scale of the solid phase as functions of the threshold for the eroded designs for η _i  = 0.4. Dashed line shows the minimum length scale for the solid phase in the dilated design for η _d  = 0.2

Fig. 21

Length scale of void phase as functions of the threshold for the dilated designs for η _i  = 0.4. Dotted line shown the minimum length scale for the void phase in the eroded design for η _e  = 0.7