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A useful dilute limit is that of spherical (circular) inclusions randomly distributed in a matrix. To be in the dilute limit, the volume fraction of the inclusion phase should probably be less than or equal to 5%, although this limit can be higher or lower, depending on inclusion shape and contrast of properties with the matrix. Call the inclusion phase 2. In general we find that, for some property being considered, say F, the property in the dilute limit has the form
Note the similar structure between these sets of formulas in 2-D and 3-D.
To test a finite difference or finite element program, one generally puts one particle in the matrix, with a unit cell-size chosen so that the volume fraction is small enough to be in the dilute limit. Even with periodic boundary conditions, the conductivity term or elastic moduli term that is linear in the volume fraction of the inclusion phase is unchanged from the infinite system limit. Refs. [11] and [12] contain electric, elastic, and viscosity dilute limits for many other shape particles.For the sphere, we use a 40 x 40 x 40 unit cell, and a 15 pixel diameter sphere, centered on the middle of a pixel, so that the sphere volume fraction is 0.028. Figure 8 shows the results for the intrinsic conductivity. The solid line is the exact solution, eq. (63). For x close to zero, it appears that the finite element is more accurate, while for x >> 1, the finite difference seems to be more accurate. To test the effect of resolution on the result, the results at x = 10, the case with the worst accuracy, were re-run for a 31 pixel diameter sphere in a 1003 unit cell. This made the volume fraction slightly smaller, which would also tend to slightly improve the results. The numbers in Table 10 in parentheses are these results. For a factor of 2 improvement in resolution, the absolute error in [σ] decreased by a factor of three for the finite element method, and by a factor of eight for the finite difference method. The result for the finite difference case only differs by 1.2% from the exact value of 2.25 in this case.
Figure 8: Intrinsic conductivity for a 15 pixel diameter sphere embedded in a
403 unit cell,
as a function of the ratio of the sphere conducticity to the matrix conductivity.
Finite element and finite difference data
and the exact result are compared.
| x | FEM | | % Error | FD | | % Error | Exact |
| 0.1 | −1.342 | 4.4 | −1.382 | 7.5 | −1.286 |
| 0.2 | −1.121 | 2.8 | −1.146 | 5.0 | −1.091 |
| 0.5 | −0.6029 | 0.5 | −0.6085 | 1.4 | −0.600 |
| 2 | 0.7653 | 2.0 | 0.7577 | 1.0 | 0.750 |
| 5 | 1.861 | 8.6 | 1.812 | 5.7 | 1.714 |
| 10 | 2.576 (2.353) | 14.5 (4.6) | 2.471 (2.278) | 9.8 (1.2) | 2.25 |
Table 10: Intrinsic conductivities for sphere, comparing the finite element and finite difference techniques. |
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For the same system, we can also calculate the intrinsic bulk and shear moduli. Choosing K1 = 1 and G1 = 0.75, and keeping, for simplicity, K2 / K1 = G2 / G1 = x (so that the same Poisson's ratio is maintained in inclusion and matrix), it turns out that using eqs. (64) and (65), both intrinsic moduli are the same and are equal to 2 (x − 1)/(x + 1). Figure 9 shows the numerical and exact results for both intrinsic moduli, for a 15 pixel diameter sphere in a 403 unit cell. The actual values are given in Table 11. The errors are qualitatively similar to the intrinsic conductivity case. The x = 10 point was rerun for the 31 pixel diameter sphere in a 1003 unit cell, and the result is shown in Table 11 in parentheses. For both the intrinsic bulk and shear moduli, the absolute error decreased by almost exactly a factor of two for the factor of two increase in resolution. There was little difference in accuracy between the computation of [K] and [G].
Figure 9: Intrinsic elastic moduli for a 15 pixel diameter sphere embedded in a
403 unit cell,
as a function of the ratio of the sphere Young's modulus to the
matrix Young's modulus.
The three sets of data show [K], [G], and the exact result, which is the same
for both intrinsic moduli.
| x | [K] | | % Error | [G] | | % Error | Exact |
| 0.1 | −1.612 | 1.5 | −1.612 | 1.5 | −1.636 |
| 0.2 | −1.31 | 1.8 | −1.328 | 0.4 | −1.333 |
| 0.5 | −0.658 | 1.3 | −0.662 | 0.7 | −0.667 |
| 2 | 0.682 | 2.3 | 0.678 | 1.7 | 0.667 |
| 5 | 1.414 | 6.0 | 1.398 | 4.9 | 1.333 |
| 10 | 1.774 (1.706) | 8.4 (4.3) | 1.755(1.697) | 7.3(3.7) | 1.636 |
Table 11: Intrinsic elastic moduli for sphere, d = 15 in 403 system |
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In 2-D (to limit computational effort), we can further analyze the resolution dependency of the finite element calculation of the intrinsic elastic moduli. A circle is placed in a computational cell, with a diameter of 1/10'th the cell size. The intrinsic moduli for the bulk and shear moduli are then computed as a function of cell size. The circle moduli are ten times larger than those of the matrix, with the same Poisson's ratio. Table 12 shows the data for this computation.
| System size | [K] | % diff. | [G] | % diff. |
| 20 | 1.548 | 10.1 | 1.318 | 6.9 |
| 40 | 1.548 | 10.1 | 1.362 | 12.9 |
| 80 | 1.475 | 4.9 | 1.290 | 4.6 |
| 160 | 1.447 | 2.9 | 1.261 | 2.3 |
| 320 | 1.433 | 1.9 | 1.251 | 1.5 |
| 640 | 1.425 | 1.3 | 1.244 | 0.9 | Table 12: Intrinsic elastic moduli for circle--effect of digital resolution, using finite element method |
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Another test of the accuracy of both the finite element and finite difference electrical programs is to run the example of the dilute limit of a cubical shaped inclusion. Since cubic pixels can replicate the shape of a cubical inclusion exactly, then any effect of a digitally rough boundary (imperfect representation of the geometry) can be dispensed with. Eyges [25] has done a very careful numerical treatment of this problem, which can be used as a check on these programs. We use a 10 x 10 x 10 cube, centered in a 40 x 40 x 40 unit cell, so that the cube volume fraction was 0.0156. The conductivity of the matrix, σ1, is taken to be unity, with the conductivity of the inclusion, σ2, ranging between 0.1 and 10. For a cubic symmetry system, the conductivity tensor is isotropic, so there is only one independent diagonal component, and no off-diagonal components. Let x = σ2 / σ1. Figure 10 shows the results. It is interesting to note that, for high values of x, the finite difference result actually tends to be a bit more accurate than does the finite element result. It is known that in the limit of very high values of x, the finite element program tends to be about 8-9% high for the dilute limit of cubes [11]. The formula used to plot Eyges' [25] data was
Figure 10: Intrinsic conductivity for a 103 cube embedded in a 403 unit cell,
as a function of the ratio of the cube conductivity to the matrix conductivity.
The three sets of data (circle, square, line)
compare the finite element method, the finite difference method, and Eyges'
data, respectively.
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