De-homogenization of optimal 2D topologies for multiple loading cases

This work presents an extension of the highly efficient de-homogenization method for obtaining high-resolution, near-optimal 2D topologies optimized for minimum compliance subjected to multiple load cases. We perform a homogenization-based topology optimization based on stiffness optimal Rank- microstructure parameterizations to obtain stiffness optimal multi-scale designs on relatively coarse grids.

To avoid relatively thin microstructure features, we regularize the design by introducing a material indicator field which results in well-defined widths and structural boundaries. In order to avoid singularities from the multiple load case problem, the orientations of the microstructures are further regularized.

Subsequently, we derive a single-scale interpretation of stiffness optimal multi-scale designs on a fine grid using de-homogenization. The single-scale interpretation can be derived without costly postprocessing analysis on the fine grid, as an implicit boundary formulation is used. The effect of starting guesses is discussed, as they are non-trivial for Rank- microstructures.

Different numerical examples verify the performance of the inexpensive high-resolution solutions, both in comparison to the Rank- based homogenization solutions, to equivalent density-based large-scale solutions, as well as to strict isotropic microstructure solutions. Depending on starting guesses, the approach consistently delivers structural performance values within a few percent of density-based large-scale solutions with a CPU time reduction factor of more than 300.

Finally, we confirm that isotropic as well as orthogonal Rank-2 microstructure models are inferior to stiffness optimal anisotropic microstructure models for minimum compliance problems subjected to multiple load cases.