lgca.multispecies_base.MultiSpeciesLGCA_base

class lgca.multispecies_base.MultiSpeciesLGCA_base(*, n_species=1, **kwargs)

Bases: LGCA_base

Classical LGCA supporting multiple species.

apply_abc()

Apply absorbing boundary conditions.

Update self.nodes, using the shadow border nodes and respecting the geometry.

apply_inflowbc()

Apply inflow boundary conditions.

Update self.nodes, using the shadow border nodes and respecting the geometry.

apply_pbc()

Apply periodic boundary conditions.

Update self.nodes, using the shadow border nodes and respecting the geometry.

apply_rbc()

Apply reflecting boundary conditions.

Update self.nodes, using the shadow border nodes and respecting the geometry.

abstract property c: ndarray

(Class attribute.) Array of the velocity channel vectors. Dimensions: (dims, lgca.velocitychannels), where dims is 1 or 2 depending on the geometry.

calc_flux(nodes)

Calculate the flux vector for all lattice sites in nodes.

The elements of the flux vectors are computed as the dot product between the LGCA’s neighborhood vectors and the velocity channel configuration in nodes.

Parameters:

nodes (numpy.ndarray) – Lattice configuration to compute the flux for. Must have more than or the same number of dimensions as self.nodes and nodes.shape[-1] >= self.velocitychannels. Is typically self.nodes.

Returns:

Array of flux vectors at each lattice site. Dimensions: nodes.shape[:-1] + (len(self.c),).

calc_permutations()

Initialize lazy computation structures for permutations. Only compute permutations when actually needed.

get_flux_permutations(n_particles)

Get flux permutations for n_particles.

get_permutations(n_particles)

Get permutations for n_particles.

Parameters:

n_particles (int) – Number of occupied channels.

Returns:

Array of permutations for n_particles.

abstractmethod gradient(qty)

Compute the gradient of qty along all axes.

Parameters:

qty (numpy.ndarray) – Quantity to take the gradient of. Needs to have the same number of dimensions as self.nodes. If qty.shape == self.nodes.shape[:-1] the result can be indexed with the LGCA coordinates (see example).

Returns:

Computed gradient. Dimensions: qty.shape + (len(self.c),). If self and qty are 2D arrays, gradient(qty)[...,0] is the gradient in x direction and gradient(qty)[...,1] the gradient in y direction.

Notes

The gradient is calculated using numpy.gradient() with stepwidth h=0.5 (s.t. no normalization takes place). It is computed as the central finite difference with equidistant support points and supports one-sided differences at the boundaries.

In most cases this yields the simple difference between the two closest array elements in the given direction. For example, the gradient at position 1 of np.array([1, 2, 4]) would be (4 - 1)/(2 * 0.5) = 3.

Examples

If the input quantity has the same x (and y) dimensions as the LGCA’s nodes, the gradient at each node position can be accessed the same way as the node itself.

>>> from lgca import get_lgca
>>> import numpy as np
>>> # define a square LGCA to illustrate dimensions
>>> lgca = get_lgca(geometry='square', dims=(2,3))
>>> lgca.nodes.shape  # (xdim, ydim, number of channels)
(4, 5, 4)
>>> my_qty = np.array([[0,0,0,0,0],
>>>                    [1,1,1,1,1],
>>>                    [2,2,2,3,2],
>>>                    [3,3,3,3,3]])
>>> my_qty.shape  # (xdim, ydim)
(4, 5)
>>> grad = lgca.gradient(my_qty)
>>> grad.shape  # (xdim, ydim, number of dimensions)
(4, 5, 2)
>>> # address like internal LGCA fields: first dimension is x (printed vertically),
>>> # second dimension is y (printed horizontally), this can be a bit confusing
>>> for coord in lgca.coord_pairs:
>>>     if np.any(grad[coord]>2):
>>>         print("Gradient at index", coord, "is ", grad[coord])
>>>         print("Configuration at index ", coord, " is ", lgca.nodes[coord],
>>>               ", with cell density ", lgca.cell_density[coord])
Gradient at index (1, 3) is  [3. 0.]
Configuration at index  (1, 3)  is  [False False False  True] , with cell density  1

The first element of the gradient holds the gradient in x direction, the second element the gradient in y direction. Note that (1, 3) is the index corresponding to a logical non-border coordinate (0, 2) if the interaction radius is 1. This is relevant for defining a custom field qty: Only the field values at non-border indices will be “felt” by the particles in the LGCA if the interaction is defined accordingly, but border nodes can be used to specify the field’s boundary conditions.

The gradient in x direction is 3 = (3 - 0)/1. In y direction it is 0 = (1 - 1)/1.

abstractmethod init_coords()

Initialize LGCA coordinates. These are used to index the lattice nodes. In the implementation, set self.nonborder, self.xcoords, self.ycoords, and self.coord_pairs to meaningful and consistent values.

Must match what is done in set_dims() and init_nodes(). For the attribute types see lgca.base.LGCA_base.

abstractmethod init_nodes(density, nodes=None, **kwargs)

Initialize LGCA lattice configuration. Create the lattice and then assign particles to channels in the nodes. In the implementation, set self.nodes.

Must match what is done in set_dims() and init_coords(). For arguments and attribute types see lgca.base.LGCA_base.

abstract property interactions: list

(Class attribute.) List of interaction functions suitable for this type of LGCA.

print_interactions()

Print the list of pre-implemented interactions for this LGCA type.

print_nodes()

Print the full lattice configuration as integers.

abstractmethod propagation()

Perform the transport step of the LGCA: Move particles through the lattice according to their velocity.

Propagate the particles by updating self.nodes, respecting the geometry. Boundary conditions are enforced later by apply_boundaries().

random_reset(density)

Randomly initialize a total density distributed over all species.

set_bc(bc)

Set the boundary conditions.

Selects a method which is called every timestep to enforce boundary conditions. The methods to select from are implemented in the derived classes. The chosen one is assigned to self.apply_boundaries().

Parameters:

bc ({'absorbing', 'reflecting', 'periodic', 'inflow'}) – Boundary conditions. Not all bc are supported in all geometries (yet).

set_dims(dims=None, nodes=None, restchannels=0, **kwargs)

Set LGCA dimensions. In the implementation, set self.K, self.restchannels and self.dims to meaningful and consistent values.

Must match what is done in init_coords() and init_nodes(). For arguments and attribute types see lgca.base.LGCA_base.

Return type:

None

set_interaction(**kwargs)

Set the interaction rule and respective needed parameters.

Set self.interaction and possibly add entries in self.interaction_params. Do not use this to specify a custom interaction. In order to do this (as of now), self.interaction and self.interaction_params must be manipulated directly from an external script.

Parameters:

• kwargs['interaction'] (str, default='random_walk') – Name of the predefined interaction in lgca.interactions.

• **kwargs – Interaction parameters.

set_r_int(r)

Change the interaction radius. Update shadow border nodes accordingly.

This has effects on self.nodes, the coordinates and the computed fields.

Parameters:

r (int) – New interaction radius.

timeevo(timesteps=100, record=False, recordN=False, recorddens=True, showprogress=True, recordpertype=False)

Perform a simulation of the LGCA for timesteps timesteps.

Different quantities can be recorded during the simulation, e.g. the total number of particles at each timestep. They are stored in LGCA attributes.

Parameters:

• timesteps (int) – How long the simulation should be performed.

• record (bool) – Record the full lattice configuration for each timestep in self.nodes_t.

• recorddens (bool) – Record the number of particles at each lattice site for each timestep in self.dens_t.

• recordN (bool) – Record the total number of particles in the lattice for each timestep in self.n_t.

• recordpertype (bool) – Record the number of particles in velocity channels/resting channels at each lattice site for each timestep in self.velcells_t and self.restcells_t, respectively.

• showprogress (bool) – Show a simple progress bar with a percentage of performed timesteps in the standard output.

Return type:

None

timestep()

Update the state of the LGCA from time k to k+1. Includes the interaction and propagation steps.

total_population()

Calculate the amount of particles in the lattice.

Returns:

Total population size.

update_dynamic_fields()

Update “fields” from the current LGCA state that store important variables to compute other dynamic steps.

Computes self.cell_density, number of particles at each lattice node.

Return type:

None

abstract property velocitychannels: int

(Class attribute.) Number of velocity channels.