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Murat Güven

Draw a unit cell

Draw a unit cell and see its stiffness and Poisson’s ratio as you draw. Can you draw one that gets wider when you stretch it?

Drag on the grid to paint material; the stiffness and Poisson’s ratio update with every stroke. Load example starts you from a cell that does get wider.

Symmetry

Tiled 3 × 3

Homogenized stiffness

CHE=[⋅⋅⋅⋅⋅⋅⋅⋅⋅]\dfrac{C^{\mathrm{H}}}{E}=\left[\begin{array}{rrr}\cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot\end{array}\right]

Properties

E1=⋅ EE_{1}=·\,E
E2=⋅ EE_{2}=·\,E
G12=⋅ EG_{12}=·\,E
ν12=⋅\nu_{12}=·
κ=⋅ E=⋅ κHS\kappa=·\,E=·\,\kappa_{\mathrm{HS}}

Computing…

Method

The drawing is one periodic unit cell of 48 × 48 bilinear elements, each solid (E = 1, ν = 0.3) or void (E = 10⁻⁹), in plane stress. Energy-based homogenization after Andreassen and Andreasen (2014) gives its effective stiffness.

Kaleidoscope and mirror drawings are solved on a quarter of the cell, which gives the same result faster. My paper also uses symmetry to reduce the domain, there for nonlinear elastic microstructures.

My paperReferencesCode

Assumptions and limits

Linear elastic, small strains, plane stress; stiffness in units of the solid’s E.

Elements that touch only at a corner still pass load in this mesh, but would be weak hinges in a real part. Give joints some width.

Cite

@software{guven_sketchpad,
  author = {Güven, Murat},
  title  = {Draw a unit cell},
  year   = {2026},
  url    = {https://likemaestro.github.io/tools/sketchpad}
}

References

  1. Andreassen, E., & Andreasen, C. S. (2014). How to determine composite material properties using numerical homogenization. Computational Materials Science, 83, 488–495. doi:10.1016/j.commatsci.2013.09.006
  2. Güven, M., & Özdemir, İ. (2026). Topology optimization of nonlinear elastic micro-structures by a domain reduction technique. Finite Elements in Analysis and Design, 260, Article 104599. doi:10.1016/j.finel.2026.104599