136
4. SIMULATING GROWTH AND FORM
Fig.4.43. The mean concentration of nutrients, the average of
both phases, around an object simulated using the accretive
growth model in which the deposition processis driven by the
combinationoflocal nutrient gradients k( c) and the amount of
contact with the environment hz( •• ) in (4.26) for a Pe number
set to the value 3.0. The color gradient blue-white indicates
a depletion of nutrients, the white-red gradient at the object
visualizes the age of the growth layers, and the oldest layersare
displayed in red.
Fig.4.44. The shape ofthe depletionzonesin the average nutrient distribution around the objectshownin Fig. 4.43, visualized
by constructing an is0 surface at a level where the average
nutrient concentration is slightlyabovezero
The mean concentration of nutrients, the average of both phases, around
an object simulated for a Pe number set to the value 3.0 is visualized in
Figs. 4.43 and 4.44. The dark blue region in Fig. 4-43 indicates regions with
a high nutrient concentration, while depleted regions are displayed in white.
The object itself is in this picture depicted in a white and red gradient, where
white sites indicate regions where layers were added on top of the object in the
most recent growth stages, while the older parts of the object are displayed
in red. In Fig. 4.43 at the left and the right of the simulated growth form areas
emerge which are depleted from nutrients. The three-dimensional shape of
these depletion zones in the average nutrient distribution around the object is
visualized in Fig. 4.44 by constructing and visualizing an isosurface at a level
where the average nutrient concentration is slightly above zero.
The results of the simulation experiments are summarized in Tables 4.4
and 4.5. Error bounds on the various measurements have been estimated by
repeating ten times the simulations at Pe :::: 0 and Pe :::: 3.0. The ratio R of the
total sink nodes to the total number of object nodes indicates the compactness
of the growth form. The fractal dimension Dt>oxof the surface of the object was
determined using a three-dimensional version of the (cube) box-counting
method described by Feder (1988).In the simulations we have determined the
number of sink nodes that have a nutrient absorption between a and a +.1a
for the objects obtained at various Peeler numbers. The number of sink nodes
with absorption rate a, N(a), can be related to a by a power law (see (4.19),
Kaandorp et al. 1996). The value of Dabs can be estimated from a log-log
plot. The exponent Dabs can be interpreted as a measure of the uniformity
of the nutrient distribution. In Tables 4-4 and 4.5 the average absorption a
in the boundary nodes and the values of D abs are listed for the various Pe
numbers. Furthermore, for each object the average x,y, z coordinate (the
center of gravity) was measured in lattice coordinates, together with the
standard deviations sd x , sdy, and sd., for respectively the x, y, z coordinates
4. SIMULATING GROWTH AND FORM
Fig.4.43. The mean concentration of nutrients, the average of
both phases, around an object simulated using the accretive
growth model in which the deposition processis driven by the
combinationoflocal nutrient gradients k( c) and the amount of
contact with the environment hz( •• ) in (4.26) for a Pe number
set to the value 3.0. The color gradient blue-white indicates
a depletion of nutrients, the white-red gradient at the object
visualizes the age of the growth layers, and the oldest layersare
displayed in red.
Fig.4.44. The shape ofthe depletionzonesin the average nutrient distribution around the objectshownin Fig. 4.43, visualized
by constructing an is0 surface at a level where the average
nutrient concentration is slightlyabovezero
The mean concentration of nutrients, the average of both phases, around
an object simulated for a Pe number set to the value 3.0 is visualized in
Figs. 4.43 and 4.44. The dark blue region in Fig. 4-43 indicates regions with
a high nutrient concentration, while depleted regions are displayed in white.
The object itself is in this picture depicted in a white and red gradient, where
white sites indicate regions where layers were added on top of the object in the
most recent growth stages, while the older parts of the object are displayed
in red. In Fig. 4.43 at the left and the right of the simulated growth form areas
emerge which are depleted from nutrients. The three-dimensional shape of
these depletion zones in the average nutrient distribution around the object is
visualized in Fig. 4.44 by constructing and visualizing an isosurface at a level
where the average nutrient concentration is slightly above zero.
The results of the simulation experiments are summarized in Tables 4.4
and 4.5. Error bounds on the various measurements have been estimated by
repeating ten times the simulations at Pe :::: 0 and Pe :::: 3.0. The ratio R of the
total sink nodes to the total number of object nodes indicates the compactness
of the growth form. The fractal dimension Dt>oxof the surface of the object was
determined using a three-dimensional version of the (cube) box-counting
method described by Feder (1988).In the simulations we have determined the
number of sink nodes that have a nutrient absorption between a and a +.1a
for the objects obtained at various Peeler numbers. The number of sink nodes
with absorption rate a, N(a), can be related to a by a power law (see (4.19),
Kaandorp et al. 1996). The value of Dabs can be estimated from a log-log
plot. The exponent Dabs can be interpreted as a measure of the uniformity
of the nutrient distribution. In Tables 4-4 and 4.5 the average absorption a
in the boundary nodes and the values of D abs are listed for the various Pe
numbers. Furthermore, for each object the average x,y, z coordinate (the
center of gravity) was measured in lattice coordinates, together with the
standard deviations sd x , sdy, and sd., for respectively the x, y, z coordinates