Temporal evolution and morphology of grain structure in three dimensions were simulated by the phase field and the Monte Carlo simulations. In order to prevent impingement of grain of like orientation, a new algorithm was adopted for both simulations. Excluding the initial stage, the average area is found to be proportional to time in the phase field and the Monte Carlo simulations. The scaled grain size and the face number distributions become time-independent in both simulations. The scaled grain size and the face number distributions obtained by the phase field simulation are in good agreement with those by the Monte Carlo method. The nearest neighbor face correlation similar to the Aboav-Weaire relation is observed in simulated grain structures by both methods. The nearest neighbor face correlation for the phase field model is quite similar to that for the Monte Carlo method. The Allen-Cahn type equation for the phase field simulation can be derived from the master equation of the Monte Carlo Model.
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