Fig. 2. Smooth and continuous microstructure connectivity.

(A) Two octahedron unit cells with different bar diameters for which connectivity is always guaranteed and a smooth transition is achieved by interpolating the bar diameter. (B) A simple cubic and a truncated octahedron unit cell for which connectivity is, in general, not guaranteed and a smooth and continuous transition is achieved by interpolating the unit cell geometry. (C) A face-x and center-x unit cell for which connectivity is only guaranteed at the unit cell boundaries and a smooth and continuous transition is achieved using hybrid unit cells composed of the two basic unit cells. The line plots show how the normalized tensile and shear moduli, E₁₁/E and G₁₂/G (in the reference frame and based on homogenized properties), vary over the transitional unit cells. Variables ${\widehat{\mathit{d}}}_{\mathit{i}}^{\text{bar}}$ and ${\hat{\mathit{v}}}_i$ are the unit cell bar diameter and volume, respectively.