Layered fields for natural tessellations on surfaces

Abstract

Mimicking natural tessellation patterns is a fascinating multi-disciplinary problem. Geometric methods aiming at reproducing such partitions on surface meshes are commonly based on the Voronoi model and its variants, and are often faced with challenging issues such as metric estimation, geometric, topological complications, and most critically, parallelization. In this paper, we introduce an alternate model which may be of value for resolving these issues. We drop the assumption that regions need to be separated by lines. Instead, we regard region boundaries as narrow bands and we model the partition as a set of smooth functions layered over the surface. Given an initial set of seeds or regions, the partition emerges as the solution of a time dependent set of partial differential equations describing concurrently evolving fronts on the surface. Our solution does not require geodesic estimation, elaborate numerical solvers, or complicated bookkeeping data structures. The cost per time-iteration is dominated by the multiplication and addition of two sparse matrices. Extension of our approach in a Lloyd's algorithm fashion can be easily achieved and the extraction of the dual mesh can be conveniently preformed in parallel through matrix algebra. As our approach relies mainly on basic linear algebra kernels, it lends itself to efficient implementation on modern graphics hardware.

Interactive Modeling of Cellular Structures on Surfaces with Application to Additive Manufacturing

“…In this work, we adopt the recent natural tessellations generation approach proposed in [ZMSS18] which models tessellation formation as a growth process initialized at a set of given seeds (sites), or initial regions, and driven by the evolution of the growing regions boundaries (interface). Starting with a set of n initial seeds over a given surface, we associate to them n evolving regions (cells).…”

“…In this work, we capitalize on a different numerical model for natural tessellations. Namely, the model proposed in [ZMSS18] which formulates tessellation formation as a growth process initialized at a set of given seeds (sites), or initial regions, thus guaranteeing a natively parallel formulation. Despite considerable performance gains, the cost is still prohibitive for interactive editing and the method is limited to static seed configurations.…”

Star-shaped metrics for mechanical metamaterial design

“…To ensure that cells are connected sets, we adopt the Voronoi growth model [Avrami 1939;Zayer et al 2018]. In this model, the Voronoi cells are defined through an isotropic growth process based on S.…”

Stable honeycomb structures and temperature based trajectory optimization for wire-arc additive manufacturing

“…Common methods for computing an approximative geodesic distance by using PDEs rely on the eikonal equation (Sethian 1996), the heat method (Crane et al 2013) or a variational interpretation (Belyaev and Fayolle 2015) built upon the heat method. A recent related work (Zayer et al 2018) is based on a growth model such that the tesselation arises as the solution of a set of time-dependent PDEs that describe concurrently evolving fronts. In this process, the computational costs depend only on the addition as well as the multiplication of two matrices.…”