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# Finite Element Computations

A recent finite element method for digital images [26] was used to generate data with which to check the results of the D-EMT. Of course, if one is going to check the D-EMT results with the finite element method results, one must ask: how accurate are the finite element method results themselves? Since these results are for concentrated, random systems, there are no exact analytical data against which to check the numerical results. Fortunately, by careful consideration of the possible sources of errors, one can establish the accuracy of the numerical data.

These numerical computations are carried out by first generating the random microstructure desired by building a digital image using square pixels on a 2-D square lattice or cubic pixels on a 3-D cubic lattice. Each pixel is then considered to be a bi-linear element (2-D) or tri-linear element (3-D) [25], so that the entire digital lattice is treated as the finite element mesh. The elastic displacements are linearly extrapolated across the pixels, which is why the the pixels are called bi-linear (2-D) and tri-linear (3-D). The elastic equations are written as a variational principle in the elastic energy, which is then minimized over the digital lattice. The effective moduli are usually defined by a stress average, although they could be defined by an energy average [25].

three main sources of error: (1) finite size effect, (2) digital resolution, and (3) statistical variation [39].

The finite size effect comes about because any given digital image, even with periodic boundary conditions, can only represent a small part of a large random solid. Here we are thinking of inclusions embedded in a matrix. There can be errors induced if the sample is not large enough to possess enough inclusions to be statistically representative. This sampling error can be assessed by running several different size samples, and seeing whether the results change between system sizes. When assessing this source of error, the inclusion size, in terms of pixels, stays the same, so that samples having larger lattices contain more inclusions.

The digital resolution error comes from using square or cubic pixels to represent the inclusions. Even if the inclusions had the same shape as the digital lattice, there would still be a resolution error since one is representing continuum equations with a digital lattice. The size of this error can be checked by holding the number of inclusions constant, and varying the size of the lattice so that there are more or fewer pixels per unit length.

The statistical variation error source simply comes about because the systems under consideration are random ones. For a given concentration of inclusions, there are many ways in which the inclusions might be randomly arranged. Each arrangement will have somewhat different elastic moduli, in general. The size of this error source can be assessed by computing the elastic moduli of several different realizations of the same system (same size lattice, same pixel size and number of inclusions).

In what follows, all the systems shown in the figures were prepared by random sequential adsorption [40], with the largest particles placed first, and then in descending order in diameter. The inner and outer particles were not allowed to overlap with any other inner or outer particle. In the placement process, a trial center was picked at random. If a particle centered at this point did not overlap any other particle, then it was placed and a new site chosen at random. If the particle did overlap another particle, then the site was abandoned and a new site was chosen at random.

Figure 2 shows the 2-D 10002 systems that were used to check the D-EMT results. There were three sizes of inclusions. On a 10002 size digital lattice, these had outer-inner diameters, in pixels, of 121-99, 91-69, and 61-39. Experience with many previous results has shown that having the diameter of the largest inclusion less than one-eighth of the size of the unit cell makes any finite size effects negligible. Holding the number of inclusions fixed to assess the digital resolution error, calculations were also made at sizes of 5002 and 20002. There was only 1 % - 2 % variation among the different sizes, so that the size used for all the runs was 10002. The statistical variation for 10002 size systems was very small, and so was neglected.

The inner particle had Young's modulus E3 = 5.0 and Poisson's ratio 3 = 0.2, the matrix had E1 = 1.0 and 1 = 0.3, while the shell had a Young's modulus E2 that ranged from 0.1 to about 10.0 and a Poisson's ratio of 2 = 0.3. Two systems were chosen, with matrix area fractions of approximately 0.3 and 0.5. Exact details of the microstructures considered are displayed in Tables 1 and 2. These details are given for future reference, as these highly accurate results for random, non-dilute systems are practically unique.

Table 1: Parameters defining the 2-D microstructures. N = number, L = large circles, M = medium circles, S = small circles, and c is area fraction. Phase 3 is the inner circle, phase 1 is the matrix, and phase 2 is the shell material.
NL NM NS c1 c2 c3
36 28 35 0.303 0.275 0.422
26 20 25 0.499 0.198 0.303

Table 2: Areas of circles and volumes of spheres used in pixels and voxels.
Circle diameter Sphere diameter Area (pixel #) Volume (voxel #)
121   11476
99   7668
91   6488
69   3720
61   2912
39   1184
28   11536
16   2176
14   1472
8   280

In 3-D, there were two systems considered. The first used only one size of composite sphere, with a ratio of outer to inner diameters of 1.75. The ratio of the unit cell size to the particle outer diameter size was chosen to be about 7, which makes finite size errors negligible. The digital resolution effect was analyzed by running the exact same geometry at sizes of 1003, 2003, and 3003. Figure 3 shows four non-consecutive parallel slices of the 3003 system. A full size range was only run for E2 = 0.1 and 10.0, which were the limits of the shell stiffness. The phase moduli were the same numerically as in 2-D. We found that at E2 = 0.1 the resolution error was essentially zero, as all three systems gave almost exactly the same answer, within less than 0.1 %. However, at E2 = 10.0, there was a 3.7 % drop in bulk modulus and 5.6 % drop in shear modulus between the 1003 and the 2003 systems. There was only a 0.5 % further drop in bulk modulus and 0.8 % in shear modulus when going to the 3003 system, so it was decided that the 2003 system would give adequate accuracy for all the shell moduli. For this size system, the outer particle diameter was 28 pixels wide, while the inner diameter was 16 pixels wide. Statistical errors between configurations were negligible.

The second 3-D system was 2003 in size, and had two size spheres, with outer and inner particle diameters of 28-16 and 14-8. Judging from the data obtained on the single-sphere results, this choice should also have given adequate accuracy, though it was not as carefully checked as were all the other systems. The detailed parameters used for both 3-D systems are presented in Tables 3 and 2. Figure 4 shows four parallel non-consecutive slices of the 2003 system used.

Table 3: Parameters defining the 3-D microstructures, which had either one or two sizes of spheres. The numbers are all for 2003 systems. N = number, L = large spheres, S = small spheres, and c is volume fraction. Phase 3 is the inner sphere, phase 1 is the matrix, and phase 2 is the shell material.
NL NS c1 c2 c3
230   0.668 0.269 0.063
220 900 0.517 0.392 0.091

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