Fig. 1. Schematic of average volume to obtain microstructure features at two points. The features Λ are defined by statistical averaging of the results of appropriately defined microscale directional solidification problems.

Fig. 2. Schematic of solution features of model M (V_M(x) and ) as applied to the domain of the problem of interest.

Fig. 3. (Left) Domain for computation (outer rectangle) and domain for performing averaging (inner rectangle). (Right) Schematic of the process for obtaining microstructure features.

Fig. 4. Schematic of the multiscale framework (steps indicated with the dark arrows).

Fig. 5. Schematic of the overall algorithm.

Fig. 6. Schematic of using interpolation to reduce the number of the required sample problem solutions. The top shows how we can use the time consuming fully-resolved model to evaluate f and Λ corresponding to an arbitrary feature. The bottom shows all obtained solution features (gray dots), a mesh generated in the feature space and how interpolation is used to obtain f and Λ based on results of sample problems corresponding to nodes of the mesh (black dots).

Fig. 7. Contour line of field t_s at value t_s = 250, 325 and 400 for one of the sample problems with V_M = 0.02281344 and discussed in the numerical examples section. Regions of t_s  250, 325 and 400 are colored with orientation angle to identify two different crystals.

Fig. 8. Contour of V_M (left two plots) and (right two plots) for temperature boundary condition T_b = 50 exp(−t/10) − 40 (the first and the third plots) and T_b = 100 exp(−t/20) − 90 (the second and fourth plots).

Fig. 9. (Left) Obtained features of model M, (V_M and ) with temperature boundary condition T_b = 50 exp(−t/10) − 40 (red square symbols) and T_b = 100 exp(−t/20) − 90 (green triangle symbols). (Right) (V_M, ) of 64 sample problems selected for applying the fully-resolved model. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 10. Obtained microstructure of 64 sample runs. Each rectangle corresponds to a feature (V_M, ) shown in the right plot of Fig. 9. The temperature gradient increases from left to right on a given row (constant growth velocity) and the growth velocity increases from bottom to top on a given column (constant temperature gradient).

Fig. 11. Obtained relation between the liquid volume fraction and temperature from sample problems with various features F_M (64 sample problems are computed but only 9 are plotted here).

Fig. 12. Predicted field of at the first, second, and third iteration for the case with T_b = 100 exp(−t/20) − 90. Contour line with value 0.7 demonstrates the predicted location of columnar to equiaxed transition (CET).

Fig. 13. Field of (the 14 microstructures shown correspond to the closest microstructure in the database). (Left) T_b = 50 exp(−t/10) − 40. (Right) T_b = 100 exp(−t/20) − 90.

Fig. 14. Field of (the 14 microstructures correspond to the closest microstructure in the database). (Left) T_b = 50 exp(−t/10) − 40. (Right) T_b = 100 exp(−t/20) − 90.

Fig. 15. Comparison of the predicted microstructures using the database approach with the microstructures obtained from solving the problem in the whole domain using the microscale model. (Left) T_b = 50 exp(−t/10) − 40. (Right) T_b = 100 exp(−t/20) − 90. For each plot (left or right): the picture in the middle is the fully-resolved result; the dark line in the middle picture is the predicted location of CET transition using the database approach ; the 14 pictures (around the middle picture) are the closest microstructure in the database based on features F_M at selected locations.

Fig. 16. (Left) Predicted temperature field and liquid volume fraction contours with values 0.95 and 0.05 at time 130 for the case with T_b = 50 exp(−t/10) − 40. (Right) Obtained microstructure (using microscale model) and liquid volume fraction contours with value 0.95 and 0.05 (using the database approach) at time 130 for the case with T_b = 50 exp(−t/10) − 40.

Fig. 17. Microstructure at time 81.6 for the case with T_b = 100 exp(−t/20) − 90 using different sampling of potential nucleation sites.

Fig. 18. (Left 3 plots) Microscopic temperature at time 81.6 for the case with T_b = 100 exp(−t/20) − 90 using different sampling of potential nucleation sites. (Right plot) Averaged microscopic temperature.

Fig. 19. Temperature field at time 81.6 for the case with T_b = 100 exp(−t/20) − 90. Contour line shows the position where the temperature is −2. (Left) Temperature field obtained from the microscale model by averaging among three computation results. (Middle) Predicted temperature field from the database approach. (Right) Predicted temperature using the Lever rule.

Fig. 20. Schematic of the solidification of an Al–Cu alloy. The very small rectangle inside the elliptic shape is used to demonstrate the relevant size of the domain of the sample problem versus the size of the domain of the problem of interest. It is magnified by 50 times in the right plot.

Fig. 21. Obtained G_M (left) and V_M (right) fields using model M. Units of axes, G_M and V_M are mm, K/μm and μm/s, respectively.

Fig. 22. (Left) Obtained solute features in the V_M and G_M coordinates. (Right) V_M and G_M for the sample runs. Units of G_M and V_M are K/μm and μm/s, respectively.

Fig. 23. Demonstration of sample problem domain with periodic boundary conditions applied at the top and bottom sides. The bottom half is the computational domain, the top half is just a copy of the solution from the bottom.

Fig. 24. Sample problem results using the fully-resolved model.

Fig. 25. Relation of volume fraction and temperature.

Fig. 26. Temperature field at time 40 s. (Left) Predicted by Lever rule. (Right) Predicted by the database approach.

Fig. 27. G_M and V_M fields (left half is result of the first iteration, right half is result after three iterations) and nearest microstructure in database at location A (95 mm, 75 mm), B (90 mm, 75 mm), C (75 mm, 75 mm), D (60 mm, 80 mm), E (90 mm, 10 mm), F (80 mm, 20 mm), G (65 mm, 35 mm), H (50 mm, 50 mm).