Conception automatique de structures lacunaires fractales

Abstract: In nature, there exist complex forms exhibiting multi-scale characteristics. Their geometry is associated with fractals, which display similar features. Additive manufacturing enables the development of new approaches for designing complex bio-inspired structures. In this thesis, we aim to model fractal lacunar structures. This raises questions regarding their representation, manipulation, and control. To address this challenge, we employ the BC-IFS (Boundary Controlled Iterated Function System) model, which allows encoding the topology of fractals. It is based on a subdivision process represented by an automaton and on incidence and adjacency constraints. However, the manipulation and exploration of fractals remain a challenge. Their design consists in imagining a self-similar shape and identifying its subdivision process in order to encode it with the BC-IFS model. Thus, it involves imagining a form from itself, which requires both skill and experience. Once the form is imagined, its subdivision process must be identified and translated using the BC-IFS model. This final step demands rigor and precision depending on the number of topological constraints.

We propose two approaches for the automatic design of fractal faces (in 2D), with a view to future extension into 3D. The first approach defines a generic subdivision process which, from input parameters, generates a family of multi-scale lacunar topologies. This process is based on defining a fractal face from its boundary, composed of fractal edges that we classify according to their topology. The second approach automatically defines a subdivision process from a crystallographic polyhedral circle packing. These packings, constructed geometrically from polyhedra, define another family of topologies. We identify a subdivision process from the initial polyhedron. In addition, we propose algorithms that translate the subdivision processes from these two approaches into topological representations using the BC-IFS model.

We exploit these two compatible approaches to design fractal structures by assembling faces. The BC-IFS model formalizes adjacency constraints at the assembly level, ensuring the topological consistency of the structures. We propose design rules within a topological editor accessible to non-specialist users of fractals.

We then introduce a measure of the topological complexity of a fractal face, based on the evolution of the number of lacunae and sub-faces across iterations. This measure converges to a limit on the attractor, which can be computed directly from the BC-IFS automaton. Finally, we propose a generalization of crystallographic polyhedral packings. The inversions used in their geometric construction correspond to a subdivision of the initial polyhedron. We explored other polyhedral subdivisions to facilitate extension to 3D in future work.

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Recommended citation: Boris Bordeaux, "Conception automatique de structures lacunaires fractales." PhD thesis, Université Bourgogne Europe, 2025