The process of glass transition during the quenching in the domain with the cold wall has been numerically simulated. We have implemented the temperature-dependent form of the previously proposed theoretical model, which combined the heat transfer in the domain and the gauge theory of glass transition, assuming the presence of topologically stable distortions (disclinations) in the forming solid. The competition between crystallization (formation of polycrystalline structure) and the formation of the amorphous disordered phase has been shown. At the relatively slow cooling rates corresponding to the formation of the crystalline phase, we observed a columnar to equiaxed transition qualitatively similar to the observed in many metallic alloys. The moving front followed the equilibrium isotherm corresponding to the equilibrium temperature of transition in the disclinations subsystem, although front drag resulted in the effect of kinetic undercooling and the emergence of the maximum velocity of the crystallization front. High thermal conductivity values associated with the substantial heat flux lead to the bulk amorphous state. The dynamics of the coarsening of the primary amorphous structure depended on the annealing temperature.