  Magnetic damping has a significant impact on the performance of various
magnetic and spintronic devices, making it a long-standing focus of research.
The strength of magnetic damping is usually quantified by the Gilbert damping
constant in the Landau-Lifshitz-Gilbert equation. Here we propose a
first-principles based approach to evaluate the Gilbert damping constant
contributed by spin-lattice coupling in magnetic insulators. The approach
involves effective Hamiltonian models and spin-lattice dynamics simulations. As
a case study, we applied our method to Y$_3$Fe$_5$O$_{12}$, MnFe$_2$O$_4$ and
Cr$_2$O$_3$. Their damping constants were calculated to be $0.8\times10^{-4}$,
$0.2\times10^{-4}$, $2.2\times 10^{-4}$, respectively at a low temperature. The
results for Y$_3$Fe$_5$O$_{12}$ and Cr$_2$O$_3$ are in good agreement with
experimental measurements, while the discrepancy in MnFe$_2$O$_4$ can be
attributed to the inhomogeneity and small band gap in real samples. The
stronger damping observed in Cr$_2$O$_3$, compared to Y$_3$Fe$_5$O$_{12}$,
essentially results from its stronger spin-lattice coupling. In addition, we
confirmed a proportional relationship between damping constants and the
temperature difference of subsystems, which had been reported in previous
studies. These successful applications suggest that our approach serves as a
promising candidate for estimating the Gilbert damping constant in magnetic
insulators.
