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Completed section on distance to boundary calculations.

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Paul Romano 2012-07-27 10:40:14 -04:00
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docs/source/methods/geometry.rst
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@ -73,20 +73,58 @@ Computing the Distance to Nearest Boundary
------------------------------------------
One of the most basic algorithms in any Monte Carlo code is determining the
distance to the nearest boundary within a cell. Since each cell is defined by
distance to the nearest surface within a cell. Since each cell is defined by
the surfaces that bound it, if we compute the distance to all surfaces bounding
a cell, we can determine the nearest one. Let us suppose we have a particle at
:math:`(x,y,z)` traveling in the direction :math:`u,v,w`. To find the distance
:math:`d` to a surface :math:`f(x,y,z) = 0`, we need to solve the equation:
a cell, we can determine the nearest one.
With the possibility of a particle having coordinates on multiple levels
(universes) in a geometry, we must exercise care when calculating the distance
to the nearest surface. Each different level of geometry has a set of boundaries
with which the particle's direction of travel may intersect. Thus, it is
necessary to check the distance to the surfaces bounding the cell in each
level. This should be done starting the highest (most global) level going down
to the lowest (most local) level. That ensures that if two surfaces on different
levels are coincident, by default the one on the higher level will be selected
as the nearest surface.
The following procedure is used to calculate the distance to each bounding
surface. Suppose we have a particle at :math:`(x,y,z)` traveling in the
direction :math:`u,v,w`. To find the distance :math:`d` to a surface
:math:`f(x,y,z) = 0`, we need to solve the equation:
.. math::
:label: dist-to-boundary-1
f(x + du, y + dv, z + dw) = 0
If the solution to equation :eq:`dist-to-boundary-1` is negative, this means
that the surface is "behind" the particle, i.e. if the particle continues
traveling in its current direction, it will not hit the surface.
If no solutions to equation :eq:`dist-to-boundary-1` exists or the only
solutions are complex, then the particle's direction of travel will not
intersect the surface. If the solution to equation :eq:`dist-to-boundary-1` is
negative, this means that the surface is "behind" the particle, i.e. if the
particle continues traveling in its current direction, it will not hit the
surface. The complete derivation for different types of surfaces used in OpenMC
will be presented in the following sections.
Once a distance has been computed to a boundary, we need to check if it is
closer than previously-computed distances to surfaces. Unfortunately, we cannot
just use the minimum function because some distances may be almost identical but
still different due to the use of floating-point arithmetic. Consequently, we
should first check for floating-point equality of the current distance
calculated and the minimum found thus far. This is done by checking if
.. math::
:label: fp-distance
\frac{| d - d_{min} |}{d_{min}} < \epsilon
where :math:`d` is the distance to a surface just calculated, :math:`d_{min}` is
the minimum distance found thus far, and :math:`\epsilon` is a small number. In
OpenMC, this parameter is set to :math:`\epsilon = 10^{-14}` since all floating
calculations are done on 8-byte floating point numbers.
Although they are not explicitly defined, it is also necessary to check the
distance to surfaces representing lattice boundaries if a lattice exists on a
given level.
Plane Perpendicular to an Axis
------------------------------