One of the important technologies for making plastic parts by an additive process is the fusion of a polymer wire that is deposited on the part under construction. The process is often referred as FDM. 3D printers use heated extrusion heads to melt the polymer. This head has a converging at its end allowing the formation of a deposited filament of the order of 400 microns while the diameter of the initial wire is of the order of 2 mm. The operating conditions must allow a good melting of the material and a good thermal homogeneity of the polymer at the exit of the printing head. Concerning the deposition of the polymer on the previous layers, the dimensions of the deposited thread, pressures, efforts on the extrusion head, temperature must be controlled to ensure good properties to the part. The aim of the work is to numerically calculate the heating of the wire from ambient to the processing temperatures, to calculate the pressure drop as a function of the speed of the wire, and to calculate the deposition process (velocity, pressure) of the polymer on the substrate. The numerical study was conducted using a 3D finite element method based on the the CIMLIB-CFD Library developed in the laboratory. The equations of fluid mechanics and heat transfer are coupled. The polymer follows a viscous non-Newtonian non-isothermal Carreau-Yasuda law. For the calculation of heating in the extrusion head, we focus our work on the operating limits of the process by determining the temperature area equal to that of the glass transition of the polymer, Tg. We show that the maximum length of the envelope below Tg can be set according to the Graetz number. The calculation makes it possible to determine the maximum possible flow rate while ensuring a satisfactory extrusion temperature. The deposition calculation makes it possible to determine the dimensions of the deposit, the forces on the extrusion head, the pressure field, as a function of the extrusion and displacement velocities of the extrusion head, gap between head and substrate and temperatures (molten polymer, substrate…). Such a calculation is a powerful tool for optimizing the deposition, avoid defects and maximize part properties.