  We present a Greens function approach to calculate the Dzyaloshinskii-Moriya
interactions (DMI) from first principles electronic structure calculations,
that is computationally more efficient and accurate than the most-commonly
employed supercell and generalized Bloch-based approaches. The method is
applied to the (111) Co/Pt bilayer where the Co- and/or Pt-thickness dependence
of the DMI coefficients are calculated. Overall, the calculated DMI are in
relatively good agreement with the corresponding values reported
experimentally. Furthermore, we investigate the effect of strain in the DMI
tensor elements and show that the isotropic N\'{e}el DMI can be significantly
modulated by the normal strains, $\epsilon_{xx},\epsilon_{yy}$ and is
relatively insensitive to the shear strain, $\epsilon_{xy}$. Moreover, we show
that anisotropic strains, $(\epsilon_{xx}-\epsilon_{yy})$ and
  $\epsilon_{xy}$, result in the emergence of anisotropic N\'{e}el- and
Bloch-type DMIs, respectively.
