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The algorithm described above has been developed specifically to be applied to images of random materials that have been generated either using a microstructure model, or by an experimental technique like x-ray tomography. Especially in the latter case, it is usually difficult or impossible to perform checks like how the results depend on the size of the image or on the resolution of the image. However, such checks can be done with model systems, especially ones which have an analytically known solution, to give an idea of what the error might be in any given computation using real images.
There are several sources of error in using this algorithm on a specific random system. The first is finite size error--does the image contain enough of the random structure so that the computed elastic moduli no longer depend on system size? The second error is, for a random system, how much do different realizations of the same size random system differ from each other? The third source of error is: how does the resolution of microstructural features affect the results? A fourth source of error, how well the minimum energy state is approximated in the solution algorithm, is much smaller, essentially on the order of the round-off error of the computer, and so is negligible.
Several series of runs were made to estimate these sources of error. The area or volume fractions of phases 1 and 2 were fixed at 0.5 each. Table 1 shows, for a fixed value of w=5 for a 2-D Gaussian system, the effect of system size at a fixed resolution on the computed elastic moduli, and their standard deviation over a 10 system average. The average values of E and change little for system size greater than 64. The standard deviation over the average of ten independent systems decreases as the system size increases.
|System size||Young's modulus||Poisson's ratio|
|32||2.408 ± 0.488||0.384 ± 0.092|
|64||2.627 ± 0.227||0.341 ± 0.028||128||2.673 ± 0.205||0.320 ± 0.039|
|256||2.633 ± 0.066||0.325 ± 0.012|
|512||2.624 ± 0.059||0.326 ± 0.013|
Table 1: The numerical results for Young's modulus and Poisson's ratio, for w=5 Gaussian 2-D systems at c1=c2=0.5, Poisson's ratio = 1/3, E1/E2=10, as a function of system size, averaged over 10 independent realizations.
Table 2 shows the effect of resolution on the computed dilute limit slopes for circles embedded in a matrix. The slopes are defined, where m is for matrix and i is for inclusion, by
|System size||MK||% difference||MG||% difference|
Table 2: Dilute limit for circle -- effect of digital resolution. The exact values are MK = 1.40625, and MG = 1.23288. The circle always had a nominal diameter one tenth that of the unit cell.
The slopes are a function of the goemetry of the inclusion and of the relative values of the four moduli . Each numerical point was for a circle whose diameter was one tenth the size of the periodic unit cell, so that there was little or no influence from the periodic boundary conditions. Increasing the system size in terms of the number of pixels per unit length also improved the resolution of the circle. Table 2 shows that a diameter of 16 pixels was sufficient to bring the computed initial slope within 3% of the theoretical value.
Table 3 shows results for a 2-D elastic checkerboard, where the unit cell contained four "checks" (two black, two white), so that the size of each check was one half the system size. By making the shear moduli equal, exact results can be obtained for all the effective moduli. In 2-D, the exact Young's modulus and Poisson's ratio are [19,20]:
|System size||K||% difference||E||% difference||Poisson's ratio||% difference|
Table 3: The numerical results for the bulk modulus, Young's modulus, and the Poisson's ratio for the 2-D elastic checkerboard with equal shear moduli (G1=G2=2, K1/K2=10), showing how the numerical result changes with digital resolution. The exact values are: K = 2.8, E = 14/3, and = 1/6.
Table 4 shows results similar to that of Table 3, but for the Gaussian system. The ratio of w to the system size is maintained, and the system size is changed, so that the resolution of individual features increases with system size. Here the same values for the individual phase moduli are used as in Table 3, so that the same oveall exact values should be obtained. If we take the individual features in the microstructure of these Gaussian images to be on the order of w, the correlation length, then the Gaussian systems obtain the same sort of accuracy in E and as does the checkerboard when comparing w to the size of one check. Having many more digitally rough interfaces in the Gaussian system does not cause the accuracy to degrade compared to the checkerboard.
|System size (w)||K||% difference||E||% difference||Poisson's ratio||% difference|
Table 4: The numerical results for the bulk modulus, Young's modulus, and Poisson's ratio, for w=5 Gaussian 2-D systems at c1=c21=G2=2, K1/K2=10, as a function of digital resolution. The exact values are: K = 2.8, E = 14/3, and = 1/6.
In 3-D, which is much more computer-time intensive, one test was run for a 643, w=5 Gaussian system with G1=G2=2, and K1/K2=10, the same numerical values as were used in 2-D. The exact values for the moduli were : K = 3.02041, E = 4.91513, and = 0.228782. The numerical results were all within 2% of these exact values, implying that the resolution was adequate and similar to that found in 2-D.
These error tests show that the system sizes and resolutions chosen to be used in the next section to compute the main results, 1282 (w=5) in 2-D and 643 (w=5) in 3-D, averaged over five independent realizations, are adequate to give roughly 5% accuracy in the results.