Team, Visitors, External Collaborators
Overall Objectives
Research Program
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Highlights of the Year
New Software and Platforms
New Results
Bilateral Contracts and Grants with Industry
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Section: New Results

Star-shaped Metrics for Mechanical Metamaterial Design

Participants : Jonàs Martínez, Mélina Skouras, Christian Schumacher, Samuel Hornus, Sylvain Lefebvre, Bernhard Thomaszewski.

Digital manufacturing technologies such as 3D printing and laser cutting enable us to fabricate designs with great geometric detail. One particular way of exploiting this capability is to create patterned sheet materials whose geometric structures can be tailored to control their macro-mechanical behavior.

A typical approach to model and analyze structured sheet materials is centered around the concept of a representative element—a tile—which is repeated, transformed, and laid out so as to generate a regular spatial tiling. Changing the shape of the representative tile allows to control macro-mechanical properties such as isotropy or negative Poisson's ratios. Generalizing this material design principle from a single representative tile to families of tiles that can be combined in a spatially-varying manner opens the door to structures with progressively-graded material properties.

At SIGGRAPH 2019 we have presented a method for designing mechanical metamaterials [14]. It is based on the novel concept of Voronoi diagrams induced by star-shaped metrics. As one of its central advantages, our approach supports interpolation between arbitrary metrics (see Figure 1). This capability opens up a rich space of tile geometries with interesting aesthetics and a wide range of mechanical properties. They include isotropic, tetragonal, orthotropic, as well as smoothly graded materials. We have validated the mechanical properties predicted by simulation through tensile tests on a set of physical prototypes. An open source C++ implementation of the technique can be found at

Figure 1. Our method generates a smoothly-graded pattern (left) when interpolating between three star-shaped distance functions (regular) on a regular honeycomb lattice. Each distance function is compactly parameterized with polar coordinates, allowing for simple interpolation in metric space as indicated by color-coding.