  In this study we discuss the performance of approximate SQS supercell models
in describing the cubic elastic properties of B1 (rocksalt)
Ti$_{0.5}$Al$_{0.5}$N alloy by using a symmetry based projection technique. We
show on the example of Ti$_{0.5}$Al$_{0.5}$N alloy, that this projection
technique can be used to align the differently shaped and sized SQS structures
for a comparison in modeling elasticity. Moreover, we focus to accurately
determine the cubic elastic constants and Zener's type elastic anisotropy of
Ti$_{0.5}$Al$_{0.5}$N. Our best supercell model, that captures accurately both
the randomness and cubic elastic symmetry, results in $C_{11}=447$ GPa,
$C_{12}=158$ GPa and $C_{44}=203$ GPa with 3% of error and $A=1.40$ for Zener's
elastic anisotropy with 6% of error. In addition, we establish the general
importance of selecting proper approximate SQS supercells with symmetry
arguments to reliably model elasticity of alloys. In general, we suggest the
calculation of nine elastic tensor elements - $C_{11}$, $C_{22}$, $C_{33}$,
$C_{12}$, $C_{13}$, $C_{23}$, $C_{44}$, $C_{55}$ and $C_{66}$, to evaluate and
analyze the performance of SQS supercells in predicting elasticity of cubic
alloys via projecting out the closest cubic approximate of the elastic tensor.
The here described methodology is general enough to be applied in discussing
elasticity of substitutional alloys with any symmetry and at arbitrary
composition.
