Page 61 - Computational Modeling in Biomedical Engineering and Medical Physics

48 Computational Modeling in Biomedical Engineering and Medical Physics

a. Elemental cell, M 0. b. First order ensemble (M 1) built with elemental
cells details are shown only for the elemental cell
in the upper right corner of the ensemble.

c. Fourth order ensemble (M 4) and its constituent d. Eighth order ensemble (M 8) and its constituents,
lower order ensembles (M 1,M 2,M 3). lower order ensembles (M 4 7).

00
Figure 2.1 The fundamental problem of volume-to-point flow; k p and k 0 are conductivities, and q
is the heat generation rate. The quantity of higher conductivity material (A 0 5 D 0 3 L 0 ), hence the
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composition of this system, and its size (A 0 5 H 0 3 L 0 ) are fixed Morega and Bejan (2005) .
ensemble shows off a minimum with respect to AR 1 5 H 1 =L 1 . The composition fac-
tor k 1 5 k p φ is actually an effective conductivity. Unlike AR 0 , which is a continuous
0
variable, AR 1 is quantized: the design recipe sets H 1 5 2L 0 , and L 1 size may vary as
multiple of 2H 0 , and the increment is, in fact a pair of mirrored elemental cells. The
next higher order already optimized ensembles are constructed by recurrently mirroring
the previously optimized construct and adding the high conductivity lane that con-
nects its mass center with the discharge port, on the boundary. The higher order
ensembles inherit the shape of the elemental cell, here a sequence of rectangles (even
order constructs) and squares (odd order constructs).
The sketchy, less natural appearance of the constructs in Fig. 2.1 is due to the
pending simplifying assumptions aimed to keep the number of optimization para-
meters to a minimum, which makes it possible to find analytical solutions to the

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The analogous problem of volume-to-point current flow was addressed in the photovoltaic power gen-
eration structures Morega and Bejan (2005).