Synthesis of Geometric Structures from Topology Optimized Multi-Scale Designs

Homogenization-based topology optimization produces multi-scale designs with optimal stiffness at a significantly reduced computational cost compared to well-established topology optimization methods. However, the homogenization approach does not directly create a mechanical structure but instead outputs parameters describing the behavior of microstructures at infinitesimally small scales. For compliance minimization, these optimal multi-scale descriptions of the optimized designs consist of lamination thicknesses and lamination orientations. The process of synthesizing high-resolution, near-optimal geometric structures from these optimal multi-scale designs is called de-homogenization. This thesis presents research on the de-homogenization of multi-scale designs obtained with homogenization-based topology optimization for compliance minimization. The thesis consists of an introductory background chapter followed by two parts about de-homogenization methods. The first part focuses on integration-based methods, while the second part proposes novel approaches that do not rely on integration. The first part further contains an investigation of microstructure orientations fields and singularities arising for single loading case (single-load) problems in two dimensions. It follows a description of a de-homogenization method for two-dimensional, singularity-containing multi-scale designs. Further, the first part contains an expansion to three dimensions of existing work for singularity-free single-load problems, achieving a reduction of computational effort of three orders of magnitude compared to density-based topology optimization. The first part concludes with a discussion of research on microstructure orientation initialization and regularization during the homogenization approach to obtain optimal multi-scale designs that are easier to de-homogenize with currently available methods. The second part contains an investigation of microstructure orientations fields and singularities arising for single-load problems in three dimensions. Then a novel approach for two- and three-dimensional de-homogenization, called the
subselection method, is presented that does not rely on integration. The subselection method is the first de-homogenization method that applies to singularity-containing three-dimensional single-load case problems with no modifications of the underlying orientation fields. The new approach precomputes a set of structural members that are locally well-aligned with the microstructure
orientations. Then an optimization chooses an evenly spaced subset resulting in a near-optimal single-scale structure. Finally, the subselection method is used to de-homogenize two-dimensional problems with multiple loading cases.