diff --git a/docs/source/_images/hex_lat.svg b/docs/source/_images/hex_lat.svg new file mode 100644 index 0000000000..cc150e711e --- /dev/null +++ b/docs/source/_images/hex_lat.svg @@ -0,0 +1,780 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/docs/source/_images/rect_lat.svg b/docs/source/_images/rect_lat.svg new file mode 100644 index 0000000000..cec6260c63 --- /dev/null +++ b/docs/source/_images/rect_lat.svg @@ -0,0 +1,711 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/docs/source/methods/geometry.rst b/docs/source/methods/geometry.rst index 1515ffa891..3439687b33 100644 --- a/docs/source/methods/geometry.rst +++ b/docs/source/methods/geometry.rst @@ -385,6 +385,101 @@ is found that contains the specified point. .. _cell-contains: +---------------------- +Finding a Lattice Tile +---------------------- + +If a particle is inside a lattice, its position inside the lattice must be +determined before assigning it to a cell. Throughout this section, the +volumetric units of the lattice will be referred to as "tiles". Tiles are +identified by thier indices, and the process of discovering which tile contains +the particle is referred to as "indexing". + +Rectilinear Lattice Indexing +---------------------------- + +Indices are assigned to tiles in a rectilinear lattice based on the tile's +position along the :math:`x`, :math:`y`, and :math:`z` axes. The figure below +maps the indecies for a 2d lattice. The indices, (1, 1), map to the +lower-left tile. (5, 1) and (5, 5) map to the lower-right and upper-right +tiles, respectively. + +.. figure:: ../_images/rect_lat.* + :align: center + :figclass: align-center + :width: 400px + + Rectilinear lattice tile indices. + +In general, a lattice tile is specified by the three indices, +:math:`(i_x, i_y, i_z)`. If a particle's current coordinates are +:math:`(x, y, z)` then the indices can be determined from these formulas: + +.. math:: + :label: rect_indexing + + i_x = \Bigg \lceil \frac{x - x_0}{p_0} \Bigg \rceil + + i_y = \Bigg \lceil \frac{y - y_0}{p_1} \Bigg \rceil + + i_z = \Bigg \lceil \frac{z - z_0}{p_2} \Bigg \rceil + +where :math:`(x_0, y_0, z_0)` are the coordinates to the lower-left-bottom +corner of the lattice, and :math:`p_0, p_1, p_2` are the pitches along the +:math:`x`, :math:`y`, and :math:`z` axes, respectively. + +Hexagonal Lattice Indexing +-------------------------- + +A skewed coordinate system is used for indexing hexagonal lattice tiles. Rather +than a :math:`y`-axis, another axis is used that is rotated 30 degrees +counter-clockwise from the :math:`y`-axis. This axis is referred to as the +:math:`\alpha`-axis. The figure below shows how 2d hexagonal tiles are mapped +with the :math:`(x, \alpha)` basis. In this system, (0, 0) maps to the center +tile, (0, 2) to the top tile, and (2, -1) to the middle tile on the right side. + +.. figure:: ../_images/hex_lat.* + :align: center + :figclass: align-center + :width: 400px + + Hexagonal lattice tile indices. + +Unfortunately, the indices cannot be determined with one simple formula as +before. Indexing requires a two-step process, a coarse step which determines a +set of 4 tiles that contains the particle and a fine step that determines which +of those 4 tiles actually contains the particle. + +In the first step, indices are found using these formulas: + +.. math:: + :label: hex_indexing + + \alpha = -\frac{x}{\sqrt{3}} + y + + i_x^* = \Bigg \lfloor \frac{x}{p_0 \sqrt{3} / 2} \Bigg \rfloor + + i_\alpha^* = \Bigg \lfloor \frac{\alpha}{p_0} \Bigg \rfloor + +where :math:`p_0` is the lattice pitch (in the :math:`x`-:math:`y` plane). The +true index of the particle could be :math:`(i_x^*, i_\alpha^*)`, +:math:`(i_x^* + 1, i_\alpha^*)`, :math:`(i_x^*, i_\alpha^* + 1)`, or +:math:`(i_x^* + 1, i_\alpha^* + 1)`. + +The second step selects the correct tile from that neighborhood of 4. OpenMC +does this by calculating the distance between the particle and the centers of +each of the 4 tiles, and then picking the closest tile. This works because +regular hexagonal tiles form a Voronoi tessellation which means that all of the +points within a tile are closest to the center of that same tile. + +Indexing along the :math:`z`-axis uses the same method from rectilinear +lattices, i.e. + +.. math:: + :label: hex_indexing_z + + i_z = \Bigg \lceil \frac{z - z_0}{p_2} \Bigg \rceil + ---------------------------------------- Determining if a Coordinate is in a Cell ---------------------------------------- diff --git a/docs/source/usersguide/input.rst b/docs/source/usersguide/input.rst index 9fe8a6708b..231cfac75d 100644 --- a/docs/source/usersguide/input.rst +++ b/docs/source/usersguide/input.rst @@ -840,7 +840,7 @@ Here is an example of a properly defined 2d rectangular lattice: ```` Element ---------------------- +------------------------- The ```` can be used to represent repeating structures (e.g. fuel pins in an assembly) or other geometry which naturally fits onto a hexagonal