On approaches for avoiding low-stiffness regions in variable thickness sheet and homogenization-based topology optimization

Abstract

Variable thickness sheet and homogenization-based topology optimization often result in spread-out, non-well-defined solutions that are difficult to interpret or de-homogenize to sensible final designs. By extensive numerical investigations, we demonstrate that such solutions are due to non-uniqueness of solutions or at least very flat minima. Much clearer and better-defined solutions may be obtained by adding a measure of non-void space to the objective function with little if any increase in structural compliance. We discuss various alternatives for cleaning up solutions and propose two efficient approaches which both introduce an auxiliary field to control non-void space: one approach based on a cut element based auxiliary field (hybrid approach) and another approach based on an auxiliary element based field (density approach). At the end, we demonstrate significant qualitative and quantitative improvements in variable thickness sheet and de-homogenization designs resulting from the proposed cleaning schemes.

Appendix: A

1.1 A.1 One-field interpolation scheme

This interpolation scheme suggested by Larsen et al. (2018) makes use of a smoothed Heaviside function (Wang et al. 2011), to obtain the physical thickness $\bar {\tilde {\mathbf {x}}}$.

$$ \bar{\tilde{\mathbf{x}}} = \tilde{\mathbf{x}}\frac{\tanh(\beta \eta) + \tanh(\beta (\tilde{\mathbf{x}}- \eta))}{\tanh(\beta \eta) + \tanh(\beta (1- \eta))}, $$

(9)

where threshold parameter η, and the sharpness of the projection β, updated each 50 iterations using the continuation approach shown in Fig. 21, where the legend indicates the order of the steps taken. This scheme results in 150 elements with $\bar {\tilde {\mathbf {x}}} \in ]0.001;\alpha _{min}[$. These values can be thresholded either to α_out or to α_min with the following scheme. First, we count the total volume contributions of these elements. Second, we identify how many elements can be set to α_min based on this volume contribution. The largest values of $\bar {\tilde {\mathbf {x}}} \in ]0.001;\alpha _{min}[$ will then be set to α_min such that the total volume is not violated. The corresponding thresholded design for this one-field interpolation scheme is shown in Fig. 22.

Fig. 21

Interpolation scheme plotted for different values of η and β, the order of the lines follows the continuation approach

Fig. 22

Optimized Michell cantilever using one-field interpolation scheme. Blue line shows boundary where thickness goes from zero to α_min