matt_papers/RSA/computational_methods.tex

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%%%%    Computational Methods
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The simulation size was 4,000 repeated hcp units
in both the x and y direction. This gave a rectangular plane, to which periodic
boundary conditions were applied. The particle's attachment point was then
randomly assigned a location on the plane. This location was then checked 
against the underlying lattice to see if they were within $\epsilon$ of 
one of the lattice gaps. If the attachment point was indeed close enough,
the particle was said to stick at that location, and the particle's new
attachment location was specified to be the lattice gap coordinate. 
All failures resulted in a new random location for the attachment point.

Once the particle was found to stick to the lattice, the particle was tested 
against the pre-existing particles for overlap. In the case of the octopi 
model, the test was a simple distance formula test. Here the centers of the
particles were specified to be at least $2\sigma$ apart. Where $\sigma$ is 
the radius of the particle. 

The test for overlap in the case of the tilted umbrella particle, is slightly
more complex. For these particles, several sequential tests are made. The
first test is the simplest, and checks to make sure that the new umbrella's
attachment point, or ``handle'', does not lie within the elliptical projection
of a previously attached umbrella's top onto the xy-plane. 
If the particle passes this first 
screening, it is then subjected to a 3-dimensional evaluation of whether the
two umbrella tops intersect. This involves using the normals of both
umbrellas, and computing the parametric line equation from the intersection
of the two planes specified by the umbrella tops. This line is
then tested for intersection with the circles defined as the umbrella tops.
If there are points of intersection, these points must be tested against 
both circles, such that the line intersects each circle sequentially. In
other words, the line must enter then leave one circle before it can enter
the next. 

To speed up the overlap tests, a modified 2-D neighbor list method was 
employed. The plane was divided into a 500 x 500 grid of equally sized
rectangular bins. The overlap test then cycled over all of the particles within
the bins located in a 3 x 3 grid centered on the bin in which the test
particle lied.
Revision:	57
Committed:	Tue Jul 30 18:47:17 2002 UTC (23 years, 3 months ago) by mmeineke
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#	User	Rev	Content
1	mmeineke	56	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2			%%%% Computational Methods
3			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
4
5			The simulation size was 4,000 repeated hcp units
6			in both the x and y direction. This gave a rectangular plane, to which periodic
7			boundary conditions were applied. The particle's attachment point was then
8			randomly assigned a location on the plane. This location was then checked
9			against the underlying lattice to see if they were within $\epsilon$ of
10			one of the lattice gaps. If the attachment point was indeed close enough,
11			the particle was said to stick at that location, and the particle's new
12			attachment location was specified to be the lattice gap coordinate.
13			All failures resulted in a new random location for the attachment point.
14
15			Once the particle was found to stick to the lattice, the particle was tested
16			against the pre-existing particles for overlap. In the case of the octopi
17			model, the test was a simple distance formula test. Here the centers of the
18			particles were specified to be at least $2\sigma$ apart. Where $\sigma$ is
19			the radius of the particle.
20
21			The test for overlap in the case of the tilted umbrella particle, is slightly
22			more complex. For these particles, several sequential tests are made. The
23			first test is the simplest, and checks to make sure that the new umbrella's
24			attachment point, or ``handle'', does not lie within the elliptical projection
25			of a previously attached umbrella's top onto the xy-plane.
26			If the particle passes this first
27			screening, it is then subjected to a 3-dimensional evaluation of whether the
28			two umbrella tops intersect. This involves using the normals of both
29			umbrellas, and computing the parametric line equation from the intersection
30			of the two planes specified by the umbrella tops. This line is
31			then tested for intersection with the circles defined as the umbrella tops.
32			If there are points of intersection, these points must be tested against
33			both circles, such that the line intersects each circle sequentially. In
34			other words, the line must enter then leave one circle before it can enter
35			the next.
36
37			To speed up the overlap tests, a modified 2-D neighbor list method was
38			employed. The plane was divided into a 500 x 500 grid of equally sized
39			rectangular bins. The overlap test then cycled over all of the particles within
40			the bins located in a 3 x 3 grid centered on the bin in which the test
41			particle lied.