  Phonons are fundamentally important for many materials properties, including
thermal and electronic transport, superconductivity, and structural stability.
Here, we describe a method to compute phonons in correlated materials using
state-of-the-art DFT+DMFT calculations. Our approach combines a robust DFT+DMFT
implementation to calculate forces with the direct method for lattice dynamics
using nondiagonal supercells. The use of nondiagonal instead of diagonal
supercells drastically reduces the computational expense associated with the
DFT+DMFT calculations. We benchmark the method for typical correlated materials
(Fe, NiO, MnO, SrVO$_3$), testing for $\mathbf{q}$-point grid convergence and
different computational parameters of the DFT+DMFT calculations. The efficiency
of the nondiagonal supercell method allows us to access $\mathbf{q}$-point
grids of up to $6\times6\times6$. In addition, we discover that for the small
displacements that atoms are subject to in the lattice dynamics calculation,
fixing the self-energy to that of the equilibrium configuration is in many
cases an excellent approximation that further reduces the cost of the DFT+DMFT
calculations. This fixed self-energy approximation is expected to hold for
materials that are not close to a phase transition. Overall, our work provides
an efficient and general method for the calculation of phonons using DFT+DMFT,
opening many possibilities for the study of lattice dynamics and associated
phenomena in correlated materials.
