  Using Cellular Automata, we simulate spin systems corresponding to $2d$ Ising
model with various kinds of boundary conditions (bcs). The appearance of
spontaneous magnetization in the absence of magnetic field is studied with a
$64\times64$ square lattice with five different bcs, i.e., periodic, adiabatic,
reflexive, fixed ($+1$ or $-1$) bcs with three initial conditions (all spins
up, all spins down and random orientation of spins). In the context of $2d$
Ising model, we have calculated the magnetisation, energy, specific heat,
susceptibility and entropy with each of the bcs and observed that the phase
transition occurs around $T_c$ = 2.269 as obtained by Onsager. We compare the
behaviour of magnetisation vs temperature for different types of bcs by
calculating the number of points close to the line of zero magnetisation after
$T>T_c$ at various lattice sizes. We observe that the periodic, adiabatic and
reflexive bcs give closer approximation to the value of $T_c$ than fixed +1 and
fixed -1 bcs with all three initial conditions for lattice size less than
$70\times70$. However, for lattice size between $70\times70$ and
$100\times100$, fixed +1 bc and fixed -1 bc give closer approximation to the
$T_c$ with initial conditions all spin down configuration and all spin up
configuration respectively.
