Fused silica is promising for layer-by-layer additive manufacturing by selective laser melting of its powder. This material is resistant to thermal shocks associated with the laser treatment. Experiments show feasibility of such a process. However, the window of acceptable process parameters seems to be very narrow. The reason is a very high viscosity of fused silica, even at elevated temperatures, because powder consolidation is controlled by viscous flow. Experimental results are analyzed with the developed mathematical model considering powder consolidation by merging softened particles. To calculate heat transfer in the consolidating powder bed, a theoretical approach to the effective thermal conductivity is proposed. The numerical modeling can predict the experimentally observed structure of the laser track. To control the quality of the obtained material, the most important process parameters are the laser power, the thickness of the powder layer, and the preheating temperature. The laser power should be sufficient to heat the surface up to the boiling point. To attain complete consolidation in the critical zone situated at the bottom of the powder layer, the thickness of the layer can be reduced. Numerical modeling indicates that another effective way to increase the temperature and the consolidation degree in the critical zone is to apply preheating. (C) 2016 Published by Elsevier Ltd.