Fig. 1. Pictographs of three different heterogeneous microstructures: On the left: pearlite; In the middle: brass; On the right: soil.

Fig. 2. Any microstructure is represented as a linear combination of the eigen-images.

Fig. 3. To make the mapping bijective, its range has to be contracted from to .

Fig. 4. Sequential contraction of to : (a) Enforcing the first-order statistics. (b) Enforcing the pixel bounds; and (c) : Enforcing one second-order statistic.

Fig. 5. Schematic of the computational system used for validation of the VMS method.

Fig. 6. Comparison of the VMS solution with the fully-resolved solution. (a) Fully-resolved simulation; (Number of elements = 262,114, δt = 1 × 10⁻²) (b)–(e) Top row, Reconstructed VMS solution; Bottom row, corresponding coarse scale VMS solutions. Number of elements used for these simulations (number within brackets is the subgrid discretization) : (b) 32,768 (8), (c) 4096 (64), (d) 512 (512), (e) 64 (4096). The coarse scale time step used in all cases: Δt = 1.0.

Fig. 7. Error between reconstructed VMS solution and fully-resolved FEM simulation with increasing coarse element size.

Fig. 8. Error between reconstructed VMS solution and fully-resolved FEM simulation with increasing coarse scale time step Δt.

Fig. 9. Experimental image of a two-phase composite (from [55]).

Fig. 10. The two-point correlation function.

Fig. 11. Comparison of the two-point correlation function from experiments and from the GRF.

Fig. 12. One instance (realization) of the two-phase microstructure.

Fig. 13. Eigen-spectrum of the reconstructed microstructural images.

Fig. 14. Steady-state mean temperature (using first-order constraints): (a) temperature contour; (b)–(d) temperature iso-surfaces; (e)–(g) temperature slices.

Fig. 15. Reduction in the interpolation error with increasing number of samples (first-order constraint imposed).

Fig. 16. Standard deviation of temperature (using first-order constraints): (a) standard deviation iso-surfaces; (b,c) temperature PDF at two points. (d)–(f) Standard deviation slices.

Fig. 17. Reduction in the interpolation error with increasing number of samples (first- and second-order constraints imposed).

Fig. 18. Steady state mean temperature (using second-order constraints): (a) temperature contour; (b)–(d) temperature iso-surfaces; (e)–(g) temperature slices.

Fig. 19. Standard deviation of temperature (using second-order constraints): (a) standard deviation iso-surfaces; (b,c) temperature PDF at two points. (d)–(f) Standard deviation slices.

Fig. 20. Comparison of temperature PDFs at two points (left, A = (22.15, 20, 0) μm and right, B = (11.69, 0, 9.23) μm) under the application of first- and second-order constraints.