Synthesis of Frame Field-Aligned Multi-Laminar Structures

In the field of topology optimization, the homogenization approach has been revived as an important alternative to the established, density-based methods. Homogenization can represent microstructures at length scales decoupled from the resolution of the computational grid. The optimal microstructure for a single load case is an orthogonal rank-3 laminate. Initially, we investigate where singularities occur in orthogonal rank-3 laminates and show that the laminar parts of the structures we seek are unaffected by the singularities. Based on this observation, we propose a method for generating multi-laminar structures from frame fields that describe rank-3 laminates. Rather than establishing a parametrization of the domain, we compute stream surfaces that align with the frame fields and solve an optimization problem in order to find a well-spaced collection of such stream surfaces. Since our method does not rely on a parametrization, we also do not need a combing of the frame fields to generate this collection. Finally, we provide a method for synthesizing multi-laminar structures from a stream surface collection. This method produces a volumetric solid for each surface and combines these to form the output. We demonstrate our method on several frame fields produced by the homogenization approach to topology optimization.