The cellular automata are models that discretize the space time and define automata in each point (cell) of the network that act in function of what their neighbors do (automata because they have a defined form of reacting). An example is a hexagonal structure:

Model D2Q7 (two dimensions and 7 elements per cell - 6 sides and 1 center)

In the case that it is applied to a particulate gas, each node may or may not contain (states 0 and 1) a particle that can only have velocities with the directions that links between cells.

In the simulation with models like cellular automata there are two phases:

- cell acts on the others
- cell processes actions of the environment

In the special case of modeling a gas the first step corresponds to streaming while the second to collision (collision).

The mathematical description is performed by the particle distribution function f(\vec{x},\vec{v},t) where \vec{x} is Position, \vec{v} the speed and t the time. As in this case there are only discrete speeds \vec{e}_i tend to indicate the distribution function as a set of functions f_i such that

(ID 8494)

El modelo D2Q9 es un modelo bidimensional (D2) en que se se conecta el nodo (punto central) en nodos a lo largo de los ejes cartesianos\\n\\nen el origen\\n\\n

$\vec{e}_0=(0,0)$

\\n\\nen las esquinas\\n\\n

$\vec{e}_1=(1,0)$

(E),\\n

$\vec{e}_2=(0,1)$

(N), \\n

$\vec{e}_3=(-1,0)$

(W) y \\n

$\vec{e}_4=(0,-1)$

(S)\\n\\ny en las diagonales\\n\\n

$\vec{e}_5=(1,1)$

(NE), \\n

$\vec{e}_6=(-1,1)$

(SE), \\n

$\vec{e}_7=(-1,-1)$

(SW) y \\n

$\vec{e}_8=(1,-1)$

(NW)

lo que se representa en la siguiente gr fica:

(ID 8496)

El modelo D2Q9 la distribuci n en equilibrio se describe mediante la ecuaci n

\\n\\nen donde los pesos de la probabilidad son\\n\\n

$\omega_0=\displaystyle\frac{4}{9}$

\\n\\n

$\omega_1=\omega_2=\omega_3=\omega_4=\displaystyle\frac{1}{9}$

\\n\\n

$\omega_5=\omega_6=\omega_7=\omega_8=\displaystyle\frac{1}{36}$

(ID 8705)

El modelo D2Q9 la distribuci n se recalcula por efecto de las colisiones mediante

(ID 8706)

The D3Q15 model is a two-dimensional model (D3) in which the node (central point) is connected in nodes along Cartesian axes\\n\\n

$(1,0,0), (-1,0,0), (0,1,0), (0,-1,0), (0,0,1) (0,0,-1)$

\\n\\nand in the corners of the bucket\\n\\n

$(1,0,1), (-1,0,1), (0,1,1) , (0,-1,1), (1,0,-1), (-1,0,-1), (0,1,-1) , (0,-1,-1)$

which is represented in the following graph:

It is easy to build models of type D3Q19 (including side edge halves) or D3Q27 (all possible points).

(ID 8497)

En la etapa de steraming (flujo) se transportan las part culas de una celda seg n la direcci n de la velocidad que tengan hacia el nodo correspondiente:

\\n\\nEn el caso del streaming se re calculan las funciones distribuci n en el sentido de que se re calculan sus componentes:\\n\\n

$f_i(\vec{x}+\vec{e}_i\delta t,t+\delta t)=f_i(\vec{x},t)$

En este punto tambi n se incluye el efecto de los bordes que en el caso est tico reflejan la velocidad (rebote) y en el caso que se est n moviendo la modifican.

(ID 8498)

En la etapa de colisiones se reasignan part culas de una direcci n de velocidad a otra:

\\n\\nPara ello es necesario modelar la colisi n que finalmente depende de la distribuci n misma y del tipo de interacci n que exista. El resultado es que la nueva funci n distribuci n ser :\\n\\n

$f_i(\vec{x},t+\delta t)=f_i(\vec{x},t)+C(f)$

(ID 8495)

El borde mas simple es una pared paralela a la red misma de modo que las part culas pueden rebotar en forma simple:

\\n\\nEn estos casos la funci n distribuci n se deja f cilmente calcular simplemente invirtiendo la velocidad en la distribuci n en la posici n sim trica al punto de rebote:\\n\\n

$f(\vec{x},e_i,t+\delta t)=f(\vec{x},-e_i,t)$

(ID 8501)

If the collision does not occur at the point of the network, but at a distance \Delta:

\\n\\nthen the function must consider the offset by weighting the contributions\\n\\n

$f_i(x_f,t+\delta t)=\displaystyle\frac{(1-\Delta)f_{-i}(x_f,t)+\Delta(f_{-i}(x_b,t)+f_{-i}(x_{f2},t)}{1+\Delta}$

(ID 8499)

If the wall shows an inclination with respect to the network it must be modeled in a more complex form:

More general edge

First, an approximate boundary must be defined to allow the necessary edge equations to be established. Then they must be applied in the process of steraming.

(ID 8500)

Si la pare se encuentra en movimiento a una velocidad $\vec{U}$ debe considerarse que puede incrementar o frenar a las part culas. En general se tendr que\\n\\n

$f_i(\vec{e}_i,t+\delta t)=f_{-i}(\vec{e}_i)+G(\vec{e}_i,\vec{U})+G(-\vec{e}_i,-\vec{U})$

\\n\\nque se puede simplificar a \\n\\n

$f_i(e_i,t+\delta t)=f_{-i}(e_i)+\displaystyle\frac{6}{c^2}(\omega_im\vec{U}\cdot\vec{e}_i)$

(ID 8502)