Procedural Voronoi Foams for Additive Manufacturing


Microstructures at the scale of tens of microns change the physical properties of objects, making them lighter or more flexible. While traditionally difficult to produce, additive manufacturing now lets us physically realize such microstructures at low cost.

In this paper we propose to study procedural, aperiodic microstructures inspired by Voronoi open-cell foams. The absence of regularity affords for a simple approach to grade the foam geometry — and thus its mechanical properties — within a target object and its surface. Rather than requiring a global optimization process, the microstructures are directly generated to exhibit a specified elastic behavior. The implicit evaluation is akin to procedural textures in computer graphics, and locally adapts to follow the elasticity field. This allows very detailed structures to be generated in large objects without having to explicitly produce a full representation — mesh or voxels — of the complete object: the structures are added on the fly, just before each object slice is manufactured.

We study the elastic behavior of the microstructures and provide a complete description of the procedure generating them. We explain how to determine the geometric parameters of the microstructures from a target elasticity, and evaluate the result on printed samples. Finally, we apply our approach to the fabrication of objects with spatially varying elasticity, including the implicit modeling of a frame following the object surface and seamlessly connecting to the microstructures.

In ACM Transactions on Graphics (TOG) — Proceedings of ACM SIGGRAPH


We would like to thank Julian Panetta for pointing out that our comparison with [Panetta et al. 2015] was using an incorrect tile. The new tile provided by the authors revealed an excellent isotropic behavior. As a result, we revised Section 5.1. In addition, we fixed an issue with Fig. 8b, and improved the accuracy of our homogenization implementation.



    author = {Mart\'{\i}nez, Jon\`{a}s and Dumas, J{\'e}r{\'e}mie and Lefebvre, Sylvain},
    title = {Procedural Voronoi Foams for Additive Manufacturing},
    journal = {ACM Trans. Graph.},
    issue_date = {July 2016},
    volume = {35},
    number = {4},
    month = jul,
    year = {2016},
    issn = {0730-0301},
    pages = {44:1--44:12},
    articleno = {44},
    numpages = {12},
    url = {},
    doi = {10.1145/2897824.2925922},
    acmid = {2925922},
    publisher = {ACM},
    address = {New York, NY, USA},
    keywords = {3D printing, additive manufacturing, material design, procedural modeling},


We would like to thank anonymous reviewers for their valuable comments, and the members of our research team for help with printing. This work was supported by ERC grant ShapeForge (StG-2012-307877).