diff --git a/docs/source/methods/geometry.rst b/docs/source/methods/geometry.rst index 4cc97e7d0..7416aee5b 100644 --- a/docs/source/methods/geometry.rst +++ b/docs/source/methods/geometry.rst @@ -617,10 +617,10 @@ condition has been applied, the particle is killed and any surface current tallies are scored to as needed. If a reflective boundary condition has been applied to the surface, surface current tallies are scored to and then the particle's direction is changed according to the procedure in :ref:`reflection`. -Note that the white boundary condition can be considered as the special case of +Note that the white boundary condition can be considered as the special case of reflective boundary condition, where the same processing method will be applied to -deal with the surface current tallies scoring, except for determining the -changes of particle's direction according to the procedures in :ref:`white`. +deal with the surface current tallies scoring, except for determining the +changes of particle's direction according to the procedures in :ref:`white`. Next, we need to determine what cell is beyond the surface in the direction of travel of the particle so that we can evaluate cross sections based on its @@ -898,15 +898,16 @@ Dxy + Eyz + Fxz + Gx + Hy + Jz + K = 0`. Thus, the gradient to the surface is .. _white: ------------------------------- +------------------------- White Boundary Conditions ------------------------------- +------------------------- -The white boundary condition is usually applied in deterministic codes, where the particle -will hit the surface and travel back with isotropic angular distribution. The change in -particle's direction is sampled from a cosine distribution instead of uniform. -Figure :num:`fig-cosine-dist` shows an example of cosine-distribution reflection on the -arbitrary surface relative to the surface normal. +The `white boundary condition `_ +is usually applied in deterministic codes, where the particle will hit the +surface and travel back with isotropic angular distribution. The change in +particle's direction is sampled from a cosine distribution instead of uniform. +Figure :num:`fig-cosine-dist` shows an example of cosine-distribution reflection +on the arbitrary surface relative to the surface normal. .. _fig-cosine-dist: @@ -914,52 +915,50 @@ arbitrary surface relative to the surface normal. :align: center :figclass: align-center - Cosine-distribution reflection on the arbitrary surface. + Cosine-distribution reflection on an arbitrary surface. + +The probability density function (pdf) for the reflected direction can be +expressed as follows, -The probability density function (pdf) for the reflected direction can be expressed -as follows, .. math:: - : label: white-reflection-pdf - - f(\mu, \phi) = frac{\mu}{\pi} d\mu d\phi = 2\mu d\mu frac{d\phi}{2\pi} - \mu \in [0, 1] - \phi \in [0, 2\pi] + :label: white-reflection-pdf + + f(\mu, \phi) d\mu d\phi = \frac{\mu}{\pi} d\mu d\phi = 2\mu d\mu \frac{d\phi}{2\pi} + +where :math:`\mu = \cos \theta` is the cosine of the polar angle between +reflected direction and the normal to the surface; and :math:`\theta` is the +azimuthal angle in :math:`[0,2\pi]`. We can separate the multivariate +probability density into two separate univariate density functions, one for +the cosine of the polar angle, -where :math:`\mu = cos(\theta)` is the cosine of the polar angle between reflected direction -and the normal to the surface; and :math:`\theta` is the azimuthal angle. -Thus, the cosine of the polar angle can extracted like this, .. math:: - : label: white-reflection-cosine - - f(\mu)d\mu = 2\mu d\mu + :label: white-reflection-cosine + f(\mu) = 2\mu -and the azimuthal angle is uniform, -.. math:: - : label: white-reflection-uniform - - f(\phi) = frac{d\phi}{2\pi} +and one for the azimuthal angle, -Then, the cosine can be sampled by analytical inversion of cumulative density distribution (cdf) -like this, .. math:: - : label: white-reflection-sqrt-prn - - \mu = sqrt{\eta_(1)} - \phi = 2\pi \eta_(2) + :label: white-reflection-uniform + + f(\phi) = \frac{1}{2\pi}. + +Each of these density functions can be sampled by analytical inversion of the +cumulative distribution distribution, resulting in the following sampling +scheme: -where :math:`\eta` is the uniform random number that is simply computed by the random number generator. -Eventually, the final reflected direction vector can be computed via the rotation of normal to -the surface like this, .. math:: - : label: white-reflection-rotation + :label: white-reflection-sqrt-prn - u^' = u \mu + frac{uw sqrt{1-\mu^2} cos(\phi) - v sqrt{1-\mu^2}sin(\phi)}{sqrt{1-w^2}} - v^' = v \mu + frac{vw sqrt{1-\mu^2} cos(phi) + u sqrt{1-\mu^2} sin(phi)}{sqrt{1-w^2}} - w^' = w \mu - sqrt{1-w^2} sqrt{1-\mu^2} cos(\phi) + \mu = \sqrt{\xi_1} \\ + \phi = 2\pi\xi_2 -The white reflection boundary can apply to any kind of surface, as long as the normal to the surface -is known as mentioned above in :ref:`reflection`. +where :math:`\xi_1` and :math:`\xi_2` are uniform random numbers on +:math:`[0,1)`. With the sampled values of :math:`\mu` and :math:`\phi`, the +final reflected direction vector can be computed via rotation of the surface +normal using the equations from :ref:`transform-coordinates`. The white boundary +condition can be applied to any kind of surface, as long as the normal to the +surface is known as in :ref:`reflection`. .. _constructive solid geometry: http://en.wikipedia.org/wiki/Constructive_solid_geometry .. _surfaces: http://en.wikipedia.org/wiki/Surface