sisl.Grid
- class sisl.Grid(shape, bc=None, sc=None, dtype=None, geometry=None)
Bases:
sisl.supercell.SuperCellChild
Real-space grid information with associated geometry.
This grid object handles cell vectors and divisions of said grid.
- Parameters
shape (float or (3,) of int) – the shape of the grid. A
float
specifies the grid spacing in Angstrom, while a list of integers specifies the exact grid size.bc (list of int (3, 2) or (3, ), optional) – the boundary conditions for each of the cell’s planes. Default to periodic BC.
sc (SuperCell, optional) – the supercell that this grid represents. sc has precedence if both
geometry
and sc has been specified. Default to[1, 1, 1]
.dtype (numpy.dtype, optional) – the data-type of the grid, default to
numpy.float64
.geometry (Geometry, optional) – associated geometry with the grid. If sc has not been passed the supercell will be taken from this geometry.
Examples
>>> grid1 = Grid(0.1, sc=10) >>> grid2 = Grid(0.1, sc=SuperCell(10)) >>> grid3 = Grid(0.1, sc=SuperCell([10] * 3)) >>> grid1 == grid2 True >>> grid1 == grid3 True >>> grid = Grid(0.1, sc=10, dtype=np.complex128) >>> grid == grid1 False
Methods
add_vacuum
(vacuum, axis)Add vacuum along the axis lattice vector
append
(other, axis)Appends other
Grid
to this grid along axisapply
(function_, *args, **kwargs)Applies a function to the grid and returns a new grid.
area
(ax0, ax1)Calculate the area spanned by the two axis ax0 and ax1
average
(axis[, weights])Average grid values along direction axis.
copy
([dtype])Copy the object, possibly changing the data-type
cross_section
(idx, axis)Takes a cross-section of the grid along axis axis
fill
(val)Fill the grid with this value
index
(coord[, axis])Find the grid index for a given coordinate (possibly only along a given lattice vector axis)
index2xyz
(index)Real-space coordinates of indices related to the grid
index_fold
(index[, unique])Converts indices from any placement to only exist in the "primary" grid
index_truncate
(index)Remove indices from outside the grid to only retain indices in the "primary" grid
interp
(shape[, order, mode])Interpolate grid values to a new grid of a different shape
Return true if all cell vectors are linearly independent
isosurface
(level[, step_size])Calculates the isosurface for a given value.
mean
(axis[, weights])Average grid values along direction axis.
mgrid
(*slices)Return a list of indices corresponding to the slices
pyamg_boundary_condition
(A, b[, bc])Attach boundary conditions to the pyamg grid-matrix A with default boundary conditions as specified for this
Grid
pyamg_fix
(A, b, pyamg_indices, value)Fix values for the stencil to value.
pyamg_index
(index)Calculate pyamg matrix indices from a list of grid indices
pyamg_source
(b, pyamg_indices, value)Fix the source term to value.
read
(sile, *args, **kwargs)Reads grid from the
Sile
using read_gridremove
(idx, axis)Removes certain indices from a specified axis.
remove_part
(idx, axis, above)Removes parts of the grid via above/below designations.
sc_index
(*args, **kwargs)set_bc
([boundary, a, b, c])Set the boundary conditions on the grid
set_boundary
([boundary, a, b, c])Set the boundary conditions on the grid
set_boundary_condition
([boundary, a, b, c])Set the boundary conditions on the grid
set_geom
(geometry)deprecated set_geom
set_geometry
(geometry)Sets the
Geometry
for the grid.set_grid
(shape[, dtype])Create the internal grid of certain size.
set_nsc
(*args, **kwargs)Set the number of super-cells in the
SuperCell
objectset_sc
(sc)Overwrites the local supercell
set_supercell
(sc)Overwrites the local supercell
smooth
([r, method, mode])Make a smoother grid by applying a filter.
sub
(idx, axis)Retains certain indices from a specified axis.
sub_part
(idx, axis, above)Retains parts of the grid via above/below designations.
sum
(axis)Sum grid values along axis axis.
swapaxes
(a, b)Swap two axes in the grid (also swaps axes in the supercell)
tile
(reps, axis)Tile grid to create a bigger one
topyamg
([dtype])Create a pyamg stencil matrix to be used in pyamg
write
(sile, *args, **kwargs)Writes grid to the
Sile
using write_gridVoxel cell size
The data-type of the grid (in str)
Data-type used in grid
Volume of the grid voxel elements
deprecated geometry
Grid shape in \(x\), \(y\), \(z\) directions
Total number of elements in the grid
- DIRICHLET = 3
- NEUMANN = 2
- OPEN = 4
- PERIODIC = 1
- add_vacuum(vacuum, axis)
Add vacuum along the axis lattice vector
- apply(function_, *args, **kwargs)[source]
Applies a function to the grid and returns a new grid.
You can also apply a function that does not return a grid (maybe you want to do some measurement). In that case, you will get the result instead of a
Grid
.- Parameters
function (str or function) – for a string the full module path to the function should be given. The function that will be called should have the grid as the first argument in its interface.
kwargs (args and) – arguments that go directly to the function call
- area(ax0, ax1)
Calculate the area spanned by the two axis ax0 and ax1
- average(axis, weights=None)[source]
Average grid values along direction axis.
- Parameters
axis (int) – unit-cell direction to average across
weights (array_like, optional) – the weights for the individual axis elements, if boolean it corresponds to 0 and 1 for false/true.
See also
numpy.average
for details regarding the weights argument
- cross_section(idx, axis)[source]
Takes a cross-section of the grid along axis axis
Remark: This API entry might change to handle arbitrary cuts via rotation of the axis
- property dcell
Voxel cell size
- property dkind
The data-type of the grid (in str)
- property dtype
Data-type used in grid
- property dvolume
Volume of the grid voxel elements
- fill(val)[source]
Fill the grid with this value
- Parameters
val (numpy.dtype) – all grid-points will have this value after execution
- property geom
deprecated geometry
- index(coord, axis=None)[source]
Find the grid index for a given coordinate (possibly only along a given lattice vector axis)
- Parameters
coord ((:, 3) or float or Shape) – the coordinate of the axis. If a float is passed axis is also required in which case it corresponds to the length along the lattice vector corresponding to axis. If a Shape a list of coordinates that fits the voxel positions are returned (all internal points also).
axis (int, optional) – the axis direction of the index, or for all axes if none.
- index2xyz(index)[source]
Real-space coordinates of indices related to the grid
- Parameters
index (array_like) – indices for grid-positions
- Returns
coordinates of the indices with respect to this grid spacing
- Return type
- index_fold(index, unique=True)[source]
Converts indices from any placement to only exist in the “primary” grid
Examples
>>> grid = Grid([10, 10, 10]) >>> assert np.all(grid.index_fold([-1, -1, -1]) == 9)
- Parameters
index (array_like) – indices for grid-positions
unique (bool, optional) – if true the returned indices are made unique after having folded the index points
- Returns
all indices are then within the shape of the grid
- Return type
See also
index_truncate
truncate indices by removing indices outside the primary cell
- index_truncate(index)[source]
Remove indices from outside the grid to only retain indices in the “primary” grid
Examples
>>> grid = Grid([10, 10, 10]) >>> assert len(grid.index_truncate([-1, -1, -1])) == 0
- Parameters
index (array_like) – indices for grid-positions
- Returns
all indices are then within the shape of the grid (others have been removed
- Return type
See also
index_fold
fold indices into the primary cell
- interp(shape, order=1, mode='wrap', **kwargs)[source]
Interpolate grid values to a new grid of a different shape
It uses the
scipy.ndimage.zoom
, which creates a finer or more spaced grid using spline interpolation.- Parameters
shape (int, array_like of len 3) – the new shape of the grid.
order (int 0-5, optional) – the order of the spline interpolation. 1 means linear, 2 quadratic, etc…
mode ({'wrap', 'mirror', 'constant', 'reflect', 'nearest'}) – determines how to compute the borders of the grid. The default is
'wrap'
, which accounts for periodic conditions.**kwargs – optional arguments passed to the interpolation algorithm The interpolation routine is
scipy.ndimage.zoom
See also
scipy.ndimage.zoom
method used for interpolation
- is_orthogonal()
Return true if all cell vectors are linearly independent
- isosurface(level, step_size=1, **kwargs)[source]
Calculates the isosurface for a given value.
It uses
skimage.measure.marching_cubes
, so you need to have scikit-image installed.- Parameters
level (float) – contour value to search for isosurfaces in the grid. If not given or None, the average of the min and max of the grid is used.
step_size (int, optional) – step size in voxels. Larger steps yield faster but coarser results. The result will always be topologically correct though.
**kwargs – optional arguments passed directly to
skimage.measure.marching_cubes
for the calculation of isosurfaces.
- Returns
numpy array of shape (V, 3) – Verts. Spatial coordinates for V unique mesh vertices.
numpy array of shape (n_faces, 3) – Faces. Define triangular faces via referencing vertex indices from verts. This algorithm specifically outputs triangles, so each face has exactly three indices.
numpy array of shape (V, 3) – Normals. The normal direction at each vertex, as calculated from the data.
numpy array of shape (V, 3) – Values. Gives a measure for the maximum value of the data in the local region near each vertex. This can be used by visualization tools to apply a colormap to the mesh.
See also
skimage.measure.marching_cubes
method used to calculate the isosurface.
- mean(axis, weights=None)
Average grid values along direction axis.
- Parameters
axis (int) – unit-cell direction to average across
weights (array_like, optional) – the weights for the individual axis elements, if boolean it corresponds to 0 and 1 for false/true.
See also
numpy.average
for details regarding the weights argument
- classmethod mgrid(*slices)[source]
Return a list of indices corresponding to the slices
The returned values are equivalent to
numpy.mgrid
but they are returned in a (:, 3) array.
- pyamg_boundary_condition(A, b, bc=None)[source]
Attach boundary conditions to the pyamg grid-matrix A with default boundary conditions as specified for this
Grid
- Parameters
A (scipy.sparse.csr_matrix) – sparse matrix describing the grid
b (numpy.ndarray) – a vector containing RHS of \(A x = b\) for the solution of the grid stencil
bc (list of BC, optional) – the specified boundary conditions. Default to the grid’s boundary conditions, else bc must be a list of elements with elements corresponding to
Grid.PERIODIC
/Grid.NEUMANN
…
- pyamg_fix(A, b, pyamg_indices, value)[source]
Fix values for the stencil to value.
- Parameters
A (
csr_matrix
/csc_matrix
) – sparse matrix describing the LHS for the linear system of equationsb (numpy.ndarray) – a vector containing RHS of \(A x = b\) for the solution of the grid stencil
pyamg_indices (list of int) – the linear pyamg matrix indices where the value of the grid is fixed. I.e. the indices should correspond to returned quantities from pyamg_indices.
value (float) – the value of the grid to fix the value at
- pyamg_index(index)[source]
Calculate pyamg matrix indices from a list of grid indices
- Parameters
index ((:, 3) of int) – a list of indices of the grid along each grid axis
- Returns
linear indices for the matrix
- Return type
See also
index
query indices from coordinates (directly passable to this method)
mgrid
Grid equivalent to
numpy.mgrid
. Grid.mgrid returns indices in shapes (:, 3), contrary to numpy’snumpy.mgrid
- Raises
ValueError – if any of the passed indices are below 0 or above the number of elements per axis
- classmethod pyamg_source(b, pyamg_indices, value)[source]
Fix the source term to value.
- Parameters
b (numpy.ndarray) – a vector containing RHS of \(A x = b\) for the solution of the grid stencil
pyamg_indices (list of int) – the linear pyamg matrix indices where the value of the grid is fixed. I.e. the indices should correspond to returned quantities from pyamg_indices.
- static read(sile, *args, **kwargs)[source]
Reads grid from the
Sile
using read_grid- Parameters
sile (Sile, str or pathlib.Path) – a
Sile
object which will be used to read the grid if it is a string it will create a new sile usingget_sile
.* (args passed directly to
read_grid(,**)
) –
- remove(idx, axis)[source]
Removes certain indices from a specified axis.
Works exactly opposite to
sub
.- Parameters
idx (array_like) – the indices of the grid axis axis to be removed
axis (int) – the axis segment from which we remove all indices idx
- remove_part(idx, axis, above)[source]
Removes parts of the grid via above/below designations.
Works exactly opposite to
sub_part
- set_bc(boundary=None, a=None, b=None, c=None)[source]
Set the boundary conditions on the grid
- Parameters
boundary ((3, 2) or (3, ) or int, optional) – boundary condition for all boundaries (or the same for all)
a (int or list of int, optional) – boundary condition for the first unit-cell vector direction
b (int or list of int, optional) – boundary condition for the second unit-cell vector direction
c (int or list of int, optional) – boundary condition for the third unit-cell vector direction
- Raises
ValueError – if specifying periodic one one boundary, so must the opposite side.
- set_boundary(boundary=None, a=None, b=None, c=None)
Set the boundary conditions on the grid
- Parameters
boundary ((3, 2) or (3, ) or int, optional) – boundary condition for all boundaries (or the same for all)
a (int or list of int, optional) – boundary condition for the first unit-cell vector direction
b (int or list of int, optional) – boundary condition for the second unit-cell vector direction
c (int or list of int, optional) – boundary condition for the third unit-cell vector direction
- Raises
ValueError – if specifying periodic one one boundary, so must the opposite side.
- set_boundary_condition(boundary=None, a=None, b=None, c=None)
Set the boundary conditions on the grid
- Parameters
boundary ((3, 2) or (3, ) or int, optional) – boundary condition for all boundaries (or the same for all)
a (int or list of int, optional) – boundary condition for the first unit-cell vector direction
b (int or list of int, optional) – boundary condition for the second unit-cell vector direction
c (int or list of int, optional) – boundary condition for the third unit-cell vector direction
- Raises
ValueError – if specifying periodic one one boundary, so must the opposite side.
- set_geometry(geometry)[source]
Sets the
Geometry
for the grid.Setting the
Geometry
for the grid is a possibility to attach atoms to the grid.It is not a necessary entity, so passing None is a viable option.
- set_nsc(*args, **kwargs)
Set the number of super-cells in the
SuperCell
objectSee
set_nsc
for allowed parameters.See also
SuperCell.set_nsc
the underlying called method
- set_sc(sc)
Overwrites the local supercell
- set_supercell(sc)
Overwrites the local supercell
- property shape
Grid shape in \(x\), \(y\), \(z\) directions
- property size
Total number of elements in the grid
- smooth(r=0.7, method='gaussian', mode='wrap', **kwargs)[source]
Make a smoother grid by applying a filter.
- Parameters
r (float or array-like of len 3, optional) –
the radius of the filter in Angstrom for each axis. If the method is
"gaussian"
, this is the standard deviation!If a single float is provided, then the same distance will be used for all axes.
method ({'gaussian', 'uniform'}) – the type of filter to apply to smoothen the grid.
mode ({'wrap', 'mirror', 'constant', 'reflect', 'nearest'}) – determines how to compute the borders of the grid. The default is wrap, which accounts for periodic conditions.
See also
- sub(idx, axis)[source]
Retains certain indices from a specified axis.
Works exactly opposite to
remove
.- Parameters
idx (array_like) – the indices of the grid axis axis to be retained
axis (int) – the axis segment from which we retain the indices idx
- sub_part(idx, axis, above)[source]
Retains parts of the grid via above/below designations.
Works exactly opposite to
remove_part
- sum(axis)[source]
Sum grid values along axis axis.
- Parameters
axis (int) – unit-cell direction to sum across
- swapaxes(a, b)[source]
Swap two axes in the grid (also swaps axes in the supercell)
If
swapaxes(0,1)
it returns the 0 in the 1 values.
- tile(reps, axis)[source]
Tile grid to create a bigger one
The atomic indices for the base Geometry will be retained.
- Parameters
See also
Geometry.tile
equivalent method for Geometry class
- topyamg(dtype=None)[source]
Create a pyamg stencil matrix to be used in pyamg
This allows retrieving the grid matrix equivalent of the real-space grid. Subsequently the returned matrix may be used in pyamg for solutions etc.
The pyamg suite is it-self a rather complicated code with many options. For details we refer to pyamg.
- Parameters
dtype (numpy.dtype, optional) – data-type used for the sparse matrix, default to use the grid data-type
- Returns
scipy.sparse.csr_matrix – the stencil for the pyamg solver
numpy.ndarray – RHS of the linear system of equations
Examples
This example proves the best method for a variety of cases in regards of the 3D Poisson problem:
>>> grid = Grid(0.01) >>> A, b = grid.topyamg() # automatically setups the current boundary conditions >>> # add terms etc. to A and/or b >>> import pyamg >>> from scipy.sparse.linalg import cg >>> ml = pyamg.aggregation.smoothed_aggregation_solver(A, max_levels=1000) >>> M = ml.aspreconditioner(cycle='W') # pre-conditioner >>> x, info = cg(A, b, tol=1e-12, M=M)
See also
pyamg_index
convert grid indices into the sparse matrix indices for
A
pyamg_fix
fixes stencil for indices and fixes the source for the RHS matrix (uses
pyamg_source
)pyamg_source
fix the RHS matrix
b
to a constant valuepyamg_boundary_condition
setup the sparse matrix
A
to given boundary conditions (called in this routine)
- write(sile, *args, **kwargs)[source]
Writes grid to the
Sile
using write_grid- Parameters
sile (Sile, str or pathlib.Path) – a
Sile
object which will be used to write the grid if it is a string it will create a new sile usingget_sile