Fig. 1. The target (left) and inverse target (right) initial distributions for solving the inverse homogenization problems.

Fig. 2. Two-dimensional base cell results (left) and corresponding periodic structures (right) for the isotropic maximum permeability problem using the (a) target and (b) inverse target initial distributions on a 60 × 60 element mesh. For reference, the center of base cell (a) contains the solid phase.

Fig. 3. Two-dimensional results for the isotropic maximum permeability problem using the (a) target and (b) inverse target initial distributions on a 100 × 100 element mesh. The results are nearly identical to Fig. 2, demonstrating mesh independence.

Fig. 4. Base cell results for the isotropic maximum permeability problem using a uniform initial distribution and prescribing velocities to be zero at the corner nodes (left), the centroid of the base cell (center), and the midpoint of the base cell boundaries (right).

Fig. 5. Periodic structures corresponding to the base cell designs in Fig. 4.

Fig. 6. The solid (left) and void/fluid (right) phases of the three-dimensional base cell results for the isotropic maximum permeability problem using the (a) target and (b) inverse target initial distributions on a 30 × 30 × 30 element mesh.

Fig. 7. The solid phase of a new base cell extracted from the periodic material that corresponds to Fig. 6b. The new base cell is nearly identical to Fig. 6a despite using different initial distributions of material.

Fig. 8. The solid (left) and void/fluid (right) phases of the three-dimensional isotropic periodic material with maximized permeability. Eight base cells are shown (2 × 2 × 2).

Fig. 9. The cross-section (left) and rotated view (right) of a Schwartz P minimal surface created using The Surface Evolver program.