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Added description for cell_contains and find_cell.

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Paul Romano 2012-07-27 12:50:15 -04:00
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docs/source/methods/geometry.rst
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@ -260,13 +260,72 @@ will then be either both positive or both negative. If they are both positive,
the smaller (closer) one will be the solution with a negative sign on the square
root of the discriminant.
----------------------------
Finding a Cell Given a Point
----------------------------
Another basic algorithm is to determine which cell contains a given point in the
global coordinate system, i.e. if the particle's position is :math:`(x,y,z)`,
what cell is it currently in. This is done in the following manner in
OpenMC. With the possibility of multiple levels of coordinates, we must perform
a recursive search for the cell. First, we start in the highest (most global)
universe which we call the base universe and do a loop over each cell within
that universe. For each cell, we check whether the specified point is inside the
cell using the algorithm described in :ref:`cell-contains`. If the cell is
filled with a normal material, the search is done and we have identified the
cell containing the point. If the cell is filled with another universe, we then
search all cells within that universe to see if any of them contain the
specified point. If the cell is filled with a lattice, the position within the
lattice is determined, and then whatever universe fills that lattice position is
recursively searched. The search ends once a cell containing a normal material
is found that contains the specified point.
.. _cell-contains:
----------------------------------------
Determining if a Coordinate is in a Cell
----------------------------------------
----------------------------
Finding a Cell Given a Point
----------------------------
One aspect of being able to determine what cell a particle is in is determining
if a particle's coordinates lie within a given cell. The current geometry
implementation in OpenMC limits all cells to being simple cells, i.e. they are
defined only with intersection of half-spaces and not unions, differences,
etc. This makes the job of determining if a point is in a cell quite simple.
The algorithm for determining if a cell contains a point is as follows. For each
surface that bounds a cell, we determine the particle's sense with respect to
the surface. As explained earlier, if we have a point :math:`(x_0,y_0,z_0)` and
a surface :math:`f(x,y,z) = 0`, the point is said to have negative sense if
:math:`f(x_0,y_0,z_0) < 0` and positive sense if :math:`f(x_0,y_0,z_0) > 0`. If
for all surfaces, the sense of the particle with respect to the surface matches
the specified sense that defines the half-space within the cell, then the point
is inside the cell.
Let us illustrate this idea with a concept. Let's say we have a cell defined as
.. code-block:: xml
<cell id="1" surfaces="-1 2 -3" />
<surface id="1" type="sphere" coeffs="0 0 0 10" />
<surface id="2" type="x-plane" coeffs="-3" />
<surface id="3" type="y-plane" coeffs="2" />
This means that the cell is defined as the intersection of the negative half
space of a sphere, the positive half-space of an x-plane, and the negative
half-space of a y-plane. Said another way, any point inside this cell must
satisfy the following equations
.. math::
:label: cell-contains-example
x^2 + y^2 + z^2 - 10^2 < 0 \\
x - (-3) > 0 \\
x - 2 < 0
So in order to determine if a point is inside the cell, we would plug its
coordinates into equation :eq:`cell-contains-example` and if the inequalities
are satisfied, than the point is indeed inside the cell.
--------------------------
Handling Surface Crossings