A Stream Surface based approach to De-homogenization in Topology Optimization

In the late 1980’s and early 1990’s Bendsøe and Kikuchi presented a computational design method today known as topology optimisation. Their work described structural designs through spatially varying laminar microstructures, proven to be optimal for mechanical compliance minimisation. The microstructures are periodic on the infinitesimal scale and the theory of homogenization was used to interpret the macroscopic material properties. Their formulation unfortunately fell out of favour by virtue of a simplified isotropic description (SIMP), which produced structures much easier to manufacture. In 2008 the homogenization was revisited with the introduction of de-homogenization,
a method to translate the infinitesimal microstructure to a manufacturable scale. This sparked a resurgence of interest and later developments have pushed the method to be an important alternative to the popular SIMP method. In this thesis we will present a novel approach to de-homogenization developed during this project. In a collaborative effort, we have constructed an approach from classical techniques for solving differential equations, optimization theory and computational geometry to solve the problem of compliance minimisation posed in mechanical engineering. In our method we trace stream surfaces through a frame field describing microstructures from homogenization-based topology optimization, we optimize the distribution of surfaces and synthesise a volumetric representation of the final structure. The structures produced exhibit high mechanical quality, rivalling the current state of the arts. This thesis additionally describes an extensive study of the guiding parameters in our de-homogenization approach. This study documents the robustness of the method and performance implications of each parameter, both in terms of mechanical quality and computational efficiency. We present an analysis of the future prospects for industrial scale topology optimization using our method. Finally we investigate the applicability of our method to hexahedral meshing. Hexahedral meshing is a very active area of investigation in computer graphics, with applications within physical simulation. We see a potential to construct an automated hex-meshing procedure form our method.