Grid¶

class sisl.Grid(shape, bc=None, sc=None, dtype=None, geometry=None)[source]¶

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

Attributes

`DIRICHLET`	Constant for defining a Dirichlet boundary condition
`NEUMANN`	Constant for defining a Neumann boundary condition
`OPEN`	Constant for defining an open boundary condition
`PERIODIC`	Constant for defining a periodic boundary condition
`cell`	Returns the inherent `SuperCell` objects `cell`
`dcell`	Returns the delta-cell
`dkind`	The data-type of the grid (in str)
`dtype`	Returns the data-type of the grid
`dvolume`	Volume of the grids voxel elements
`geom`
`icell`	Returns the inherent `SuperCell` objects `icell`
`isc_off`	Returns the inherent `SuperCell` objects `isc_off`
`n_s`	Returns the inherent `SuperCell` objects `n_s`
`nsc`	Returns the inherent `SuperCell` objects `nsc`
`origo`	Returns the inherent `SuperCell` objects `origo`
`rcell`	Returns the inherent `SuperCell` objects `rcell`
`sc_off`	Returns the inherent `SuperCell` objects `sc_off`
`shape`	Returns the shape of the grid
`size`	Returns size of the grid
`volume`	Returns the inherent `SuperCell` objects vol

Methods

`__init__`(shape[, bc, sc, dtype, geometry])	Initialize self.
`add_vacuum`(vacuum, axis)	Add vacuum along the axis lattice vector
`append`(other, axis)	Appends other `Grid` to this grid along axis
`average`(axis[, weights])	Returns the average grid along direction axis.
`copy`()	Returns a copy of the object.
`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])	Returns the index along axis axis where coord exists
`index2xyz`(index)	Real-space coordinates of indices related to the grid
`interp`(shape[, method])	Returns an interpolated version of the grid
`is_orthogonal`()	Return true if all cell vectors are linearly independent
`mean`(axis[, weights])	Returns the average grid 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_grid
`remove`(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)	Call local `SuperCell` object `sc_index` function
`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)	Sets the `Geometry` for the grid.
`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` object
`set_sc`(sc)	Overwrites the local supercell
`set_supercell`(sc)	Overwrites the local supercell
`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)	Returns the grid summed along axis axis.
`swapaxes`(a, b)	Returns Grid with swapped axis
`topyamg`()	Create a pyamg stencil matrix to be used in pyamg
`write`(sile, args, *kwargs)	Writes grid to the `Sile` using write_grid

DIRICHLET = 3¶: Constant for defining a Dirichlet boundary condition

NEUMANN = 2¶: Constant for defining a Neumann boundary condition

OPEN = 4¶: Constant for defining an open boundary condition

PERIODIC = 1¶: Constant for defining a periodic boundary condition

add_vacuum(vacuum, axis)¶

Add vacuum along the axis lattice vector

Parameters:	vacuum : float amount of vacuum added, in Ang axis : int the lattice vector to add vacuum along

append(other, axis)[source]¶: Appends other Grid to this grid along axis

average(axis, weights=None)[source]¶

Returns the average grid along direction axis.

Parameters:	axis : int unit-cell direction to average across weights : array_like 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

cell¶: Returns the inherent SuperCell objects cell

copy()[source]¶: Returns a copy of the object.

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

dcell¶: Returns the delta-cell

dkind¶: The data-type of the grid (in str)

dtype¶: Returns the data-type of the grid

dvolume¶: Volume of the grids voxel elements

fill(val)[source]¶

Fill the grid with this value

Parameters:	val : numpy.dtype all grid-points will have this value after execution

geom¶

icell¶: Returns the inherent SuperCell objects icell

index(coord, axis=None)[source]¶

Returns the index along axis axis where coord exists

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 the axis direction of the index

index2xyz(index)[source]¶

Real-space coordinates of indices related to the grid

Parameters:	index : array_like indices for grid-positions
Returns:	numpy.ndarray: coordinates of the indices with respect to this grid spacing

interp(shape, method='linear', **kwargs)[source]¶

Returns an interpolated version of the grid

Parameters:	shape : int, array_like the new shape of the grid method : str the method used to perform the interpolation, see `scipy.interpolate.interpn` for further details. **kwargs : optional arguments passed to the interpolation algorithm The interpolation routine is `scipy.interpolate.interpn`

is_orthogonal()¶: Return true if all cell vectors are linearly independent

isc_off¶: Returns the inherent SuperCell objects isc_off

mean(axis, weights=None)¶

Returns the average grid along direction axis.

Parameters:	axis : int unit-cell direction to average across weights : array_like 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.

Parameters:	*slices : slice or list of int or int return a linear list of indices that points to the collective slice made by the passed arguments
Returns:	indices : (:, 3), linear indices for each of the sliced values

n_s¶: Returns the inherent SuperCell objects n_s

nsc¶: Returns the inherent SuperCell objects nsc

origo¶: Returns the inherent SuperCell objects origo

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 : scipy.sparse.csr_matrix: sparse matrix describing the LHS for the linear system of equations
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.
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:	pyamg linear indices for the matrix
Raises:	ValueError : if any of the passed indices are below 0 or above the number of elements per axis

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’s numpy.mgrid

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.

rcell¶: Returns the inherent SuperCell objects rcell

static read(sile, *args, **kwargs)[source]¶

Reads grid from the Sile using read_grid

Parameters:	sile : Sile, str a `Sile` object which will be used to read the grid if it is a string it will create a new sile using `get_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

Parameters:	idx : int the index of the grid axis axis to be removed for `above=True` grid[:idx,…] for `above=False` grid[idx:,…] axis : int the axis segment from which we retain the indices idx above: bool if `True` will retain the grid: `grid[:idx,...]` else it will retain the grid: `grid[idx:,...]`

sc_index(*args, **kwargs)¶: Call local SuperCell object sc_index function

sc_off¶: Returns the inherent SuperCell objects sc_off

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_geom(geometry)¶

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_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_grid(shape, dtype=None)[source]¶: Create the internal grid of certain size.

set_nsc(*args, **kwargs)¶

Set the number of super-cells in the SuperCell object

See 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

shape¶: Returns the shape of the grid

size¶: Returns size of the grid

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
Raises:	ValueError : if the length of the indices is 0

sub_part(idx, axis, above)[source]¶

Retains parts of the grid via above/below designations.

Works exactly opposite to remove_part

Parameters:	idx : int the index of the grid axis axis to be retained for `above=True` grid[idx:,…] for `above=False` grid[:idx,…] axis : int the axis segment from which we retain the indices idx above: bool if `True` will retain the grid: `grid[idx:,...]` else it will retain the grid: `grid[:idx,...]`

sum(axis)[source]¶

Returns the grid summed along axis axis.

Parameters:	axis : int unit-cell direction to sum across

swapaxes(a, b)[source]¶

Returns Grid with swapped axis

If swapaxes(0,1) it returns the 0 in the 1 values.

topyamg()[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.

Returns:	A : scipy.sparse.csr_matrix which contains the grid stencil for a pyamg solver. b : numpy.ndarray containing RHS of the linear system of equations.

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 value
pyamg_boundary_condition: setup the sparse matrix A to given boundary conditions (called in this routine)

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)

volume¶: Returns the inherent SuperCell objects vol

write(sile, *args, **kwargs)[source]¶

Writes grid to the Sile using write_grid

Parameters:	sile : Sile, str a `Sile` object which will be used to write the grid if it is a string it will create a new sile using `get_sile`