4.6. ACCRETIVE GROWTH
nating from left to right and from right to left in Fig. 4.38. Initially the flow
is directed from left to right. The flow pattern about the obstacle (substrate
and growth form) is determined in the lattice Boltzmann iteration, using
ten iterat ion steps followed by a tra cer step. In each growth step ten iteration steps are used to ensure that an equilibrium is obtained in the flow
pattern. In the last step tracer particles are released from the source plane,
the lattice sites located in the xz-plane at y = ymax. The tracer particles are
absorbed by the fluid nodes adjacent to obstacle nodes; these can be either
nodes in the substrate plane (the xz-plane at y = 1) or nodes representing
the surface of the simulated growth form. The nodes representing the object are indicated as object nodes , while the object nodes at the surface of
the object are indicated as sink nodes. In this growth model it is assumed
that both the tracer distribution and flow velocities are in equilibrium and
the growth velocity of the growth form is much slower than the dispersion
of the tracer. In the sink nod es the amount of absorb ed tracer particles is
determined. After each growth step the geometrical object is mapped again
onto the thr ee-dimensional lattice with 144 3 sites. In each growth step both
the flowcomputation and the tracer step are done in two phases: left-to-right
iphaseii and right -to-left (phaseu, The nutrient grad ients k1(c) and k 2 (c),
along the mean surface normal vector, are estimated for both phases. The
final estimat ion of the nutrient grad ient k(c) is the average of k1(c) and k 2( c).
The accretive growth model in an alternating flow can be summarized
in algorithmic form, using the pseudo-code below:
initialize growth f orm
initialize f low direction
initialize t r a c e r dis tributions
do {
-Ma p the geomet rical object onto t h e l a t t i c e mo d el ;
- Ph a s e : fl o w from l e f t to r ight {
-compute f l o w veloci ties unti l e q u il ibrium ;
-comp ute tracer d i st ri but i on u ntil e q u i l ibrium;
- d e t e r mine l o c al n u t r i e n t g r ad i ent k1(c) along
n o rmal v ector; }
- Phase : fl ow f r om r i g h t to le ft {
-compute fl o w ve loci t ies until e q u i l i b r i um;
-comp u t e t r a c er distributi on until equ i l i b r i u m;
-de t e rmi ne l o c al nutri ent gradient k2 (c) a long
n ormal vector ; }
- c o mp u t e local mean nut ri e nt gra dient
k(c) = !(k1(c) + k2 (c)) ;
2
-compute l o c al amount of con tact with environment
h2 Uow_norm_cur v, av_norm_curv) a t t h e g rowth form;
- c o mp u t e the local thickness I o f a n e w
layer usi ng ei t her (4 .2 5) or (4.26 );
- Co n s t r u c t a new l a y e r on top of the
the previous layer , insert new t r i a n gl e s at
expanding sites, delete triangles at shrinking
sites ;
-D o a collisi on detection test, inhibit growth
in sites which tend to grow through
sites representing the growth form;
} until ready
In Fig.4.39 the morphologies of objects resulting from the accretive
growth model are shown for a deposit ion process exclusivelydriven by local
nutrient gradients k( c) (4.25) and various Peeler numbers. In Fig. 4.40 slices
133
nating from left to right and from right to left in Fig. 4.38. Initially the flow
is directed from left to right. The flow pattern about the obstacle (substrate
and growth form) is determined in the lattice Boltzmann iteration, using
ten iterat ion steps followed by a tra cer step. In each growth step ten iteration steps are used to ensure that an equilibrium is obtained in the flow
pattern. In the last step tracer particles are released from the source plane,
the lattice sites located in the xz-plane at y = ymax. The tracer particles are
absorbed by the fluid nodes adjacent to obstacle nodes; these can be either
nodes in the substrate plane (the xz-plane at y = 1) or nodes representing
the surface of the simulated growth form. The nodes representing the object are indicated as object nodes , while the object nodes at the surface of
the object are indicated as sink nodes. In this growth model it is assumed
that both the tracer distribution and flow velocities are in equilibrium and
the growth velocity of the growth form is much slower than the dispersion
of the tracer. In the sink nod es the amount of absorb ed tracer particles is
determined. After each growth step the geometrical object is mapped again
onto the thr ee-dimensional lattice with 144 3 sites. In each growth step both
the flowcomputation and the tracer step are done in two phases: left-to-right
iphaseii and right -to-left (phaseu, The nutrient grad ients k1(c) and k 2 (c),
along the mean surface normal vector, are estimated for both phases. The
final estimat ion of the nutrient grad ient k(c) is the average of k1(c) and k 2( c).
The accretive growth model in an alternating flow can be summarized
in algorithmic form, using the pseudo-code below:
initialize growth f orm
initialize f low direction
initialize t r a c e r dis tributions
do {
-Ma p the geomet rical object onto t h e l a t t i c e mo d el ;
- Ph a s e : fl o w from l e f t to r ight {
-compute f l o w veloci ties unti l e q u il ibrium ;
-comp ute tracer d i st ri but i on u ntil e q u i l ibrium;
- d e t e r mine l o c al n u t r i e n t g r ad i ent k1(c) along
n o rmal v ector; }
- Phase : fl ow f r om r i g h t to le ft {
-compute fl o w ve loci t ies until e q u i l i b r i um;
-comp u t e t r a c er distributi on until equ i l i b r i u m;
-de t e rmi ne l o c al nutri ent gradient k2 (c) a long
n ormal vector ; }
- c o mp u t e local mean nut ri e nt gra dient
k(c) = !(k1(c) + k2 (c)) ;
2
-compute l o c al amount of con tact with environment
h2 Uow_norm_cur v, av_norm_curv) a t t h e g rowth form;
- c o mp u t e the local thickness I o f a n e w
layer usi ng ei t her (4 .2 5) or (4.26 );
- Co n s t r u c t a new l a y e r on top of the
the previous layer , insert new t r i a n gl e s at
expanding sites, delete triangles at shrinking
sites ;
-D o a collisi on detection test, inhibit growth
in sites which tend to grow through
sites representing the growth form;
} until ready
In Fig.4.39 the morphologies of objects resulting from the accretive
growth model are shown for a deposit ion process exclusivelydriven by local
nutrient gradients k( c) (4.25) and various Peeler numbers. In Fig. 4.40 slices
133