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- Figure 1: Graphic view of cubic pixel = tri-linear finite element, showing the
1-8 labels of the vertices. The i,j,k axes coincide with the x,y,z axes.
- Figure 2: Schematic view of the structure of the Hessian matrix when thermal
strains are used (u stands for elastic displacements, ε represents
the macrostrains).
- Figure 3: Illustration of nodes near a boundary in the finite difference method.
- Figure 4: Illustration of how periodic boundary conditions are implemented
in the finite difference programs.
- Figure 5: Showing σ vs. σ2 when σ2 differs only
slightly from σ1. The points are finite element data, and the straight
line is Brown's exact expansion to second order in the
contrast (σ2 − σ1)
.
- Figure 6: Showing the
checkerboard microstructure, with dark gray being phase 2 and light gray phase 1.
- Figure 7: Showing the effective conductivity and average of the electric field magnitude
squared in phase 2 for the checkerboard, as a function of σ2. The points
are numerical finite element results, and the lines are the
exact results discussed in the
text. The system size was 128 x 128.
- 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.
- 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.
- 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.
- Figure 11: Thermal stress for
inclusions with the same elastic moduli as the matrix, but different
thermal eigenstrain. Stress is high (top)
to low (bottom) in color bar).
Left: − σxx, right: negative of the trace of the
stress tensor. Images from top:
Ellipse, 5-pointed star, 6-pointed star.
- Figure 12: Showing the resistance between two nodes, separated by 20 lattice
spacings, of a simple cubic network,
normalized by the resistance of one bond, as a function of the thickness of the
network.
- Figure 13: Image of the horizontal current magnitudes, with the applied electric field
in the x-direction in all images. The inclusion is phase 2.
Left: σ2 = 10,
right: σ2 = 0.1, and both images had σ1 = 1.0.
Top: Finite element solution. Middle: finite difference solution.
Bottom: exact solution, no periodic boundary conditions.
Color bar shows high (red) to low (black) current scale.
- Figure 14: Current distribution for the same circular inclusion problem as in
Fig. 13,
calculated by all three methods (finite element, finite difference, and exact
calculation, σ1 = 1.0, σ2 = 10.
- Figure 15: Current distribution for the same circular inclusion problem as
in Fig. 13,
calculated by all three methods (finite element, finite difference, and exact
calculation, σ1 = 1.0, σ2 = 0.1
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