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Today's photovoltaic market is divided into multicrystalline silicon wafers from ingot casting and monocrystalline wafers from Czochralski crystals. In both cases large crystals have to be cut into thin wafer geometry. As an alternative approach the EFG process and the string ribbon process are in industrial use for the production of silicon ribbons directly out of the melt. One future alternative for a high production process is the Ribbon Growth on Substrate (RGS) process which is now in the final development stage for industrial production of silicon ribbons. In this paper we will report simulation results for the crystallisation of silicon melt in contact with a substrate. This is the basis not only for the RGS process but also for thin film processes like laser remelting of amorphous Si. The results show the general solidification behaviour at the tip region of a growing Si sheet. Different crystal growth modes are predicted resulting in different grain structures of the silicon sheet as observed experimentally.