320
S. Ghosh et al.
Fig. 16 (a) Convergence with respect to the L 2 -norm of error with progressive iterations and (b)
map of error in the S 2 (r, θ ) field between the experimental MVE and reconstructed SE-MVEs and
(c) square window containing the final SE-MVE microstructure with N f ∼ 2000 fibers
in Fig. 15. The process initializes the positions of N f chosen fibers in hexagonal
close packing arrangements within the MVE, with the initial spacing between
fibers determined from the global volume fraction S 1 . The radii of these fibers are
assigned by randomly sampling from the fiber size distribution in Fig. 15c. Values
of parameters used to start the process are N f ≈ 2000, S 1 = 0.65, mean fiber
radius = 2.43 μm, and standard deviation of = 0.11 μm. The reconstruction process
executes a large number of iterations (∼1000N f ). The centroid of a randomly
selected fiber is perturbed in each iteration to transform the hexagonal arrangement
to an amorphous microstructure. Nonoverlapping perturbations are accepted for
the first ∼60% of the iterations toward this amorphization. The subsequent 40%
iterations optimize the configuration of the SE-MVE by minimizing the L 2 -norm of
the difference between the SE-MVE instantiation and the reference MVE.
||S
exp
2 (r, θ ) − S
SEMV E
2
(r, θ )|| L 2 =
MV E
S
exp
2 (r, θ ) − S SEMV E
2
(r, θ )
2 dd
(28)
After each nonoverlapping fiber shuffle, S 2 (r, θ ) of the SE-MVE is evaluated,
and the resulting L 2 -norm is compared with that for the unperturbed state. The
move is accepted only if there is an improvement with respect to the L 2 -norm.
Figure 16a shows the convergence in S 2 (r, θ ) as a function of iterations. The rate of
convergence is rapid in the beginning of the amorphization. At ∼60% of iterations,
the optimization algorithm begins with a sharp increase in convergence rate and
stabilizes at around 70% of the iterations. The map of error in the S 2 (r, θ ) field,
between the experimental MVE and the reconstructed final SE-MVE of Fig. 16c,
is given in Fig. 16b. Figure 16c shows the square window containing the SEMVE
represented by the fibers.
S. Ghosh et al.
Fig. 16 (a) Convergence with respect to the L 2 -norm of error with progressive iterations and (b)
map of error in the S 2 (r, θ ) field between the experimental MVE and reconstructed SE-MVEs and
(c) square window containing the final SE-MVE microstructure with N f ∼ 2000 fibers
in Fig. 15. The process initializes the positions of N f chosen fibers in hexagonal
close packing arrangements within the MVE, with the initial spacing between
fibers determined from the global volume fraction S 1 . The radii of these fibers are
assigned by randomly sampling from the fiber size distribution in Fig. 15c. Values
of parameters used to start the process are N f ≈ 2000, S 1 = 0.65, mean fiber
radius = 2.43 μm, and standard deviation of = 0.11 μm. The reconstruction process
executes a large number of iterations (∼1000N f ). The centroid of a randomly
selected fiber is perturbed in each iteration to transform the hexagonal arrangement
to an amorphous microstructure. Nonoverlapping perturbations are accepted for
the first ∼60% of the iterations toward this amorphization. The subsequent 40%
iterations optimize the configuration of the SE-MVE by minimizing the L 2 -norm of
the difference between the SE-MVE instantiation and the reference MVE.
||S
exp
2 (r, θ ) − S
SEMV E
2
(r, θ )|| L 2 =
MV E
S
exp
2 (r, θ ) − S SEMV E
2
(r, θ )
2 dd
(28)
After each nonoverlapping fiber shuffle, S 2 (r, θ ) of the SE-MVE is evaluated,
and the resulting L 2 -norm is compared with that for the unperturbed state. The
move is accepted only if there is an improvement with respect to the L 2 -norm.
Figure 16a shows the convergence in S 2 (r, θ ) as a function of iterations. The rate of
convergence is rapid in the beginning of the amorphization. At ∼60% of iterations,
the optimization algorithm begins with a sharp increase in convergence rate and
stabilizes at around 70% of the iterations. The map of error in the S 2 (r, θ ) field,
between the experimental MVE and the reconstructed final SE-MVE of Fig. 16c,
is given in Fig. 16b. Figure 16c shows the square window containing the SEMVE
represented by the fibers.