In this work, a numerical study of the fused deposition modeling (FDM) process is performed. The FDM process, well-known in 3D printing applications, is a popular fabrication process used in rapid prototyping and manufacturing functional parts. Due to the complexity of simulating the process, it is common to focus the study on a single strand extrusion. In particular, we are trying to improve the FDM simulation by adding the effect of neighboring printed material (the underneath layer and the adjacent previously printed strand) to modeling single filaments. Thereby, important information about the resultant printed strand, such as the cross-section and the contact interface between filaments, can be predicted. An accurate prediction of these quantities is crucial in an FDM process since they directly impact the final object's mechanical properties. The simulation framework is based on the stabilized finite element method. A boundary-conforming free-surface approach is used to track the filament border. Therefore, the computational domain matches exactly the filament shape. In general, boundary-conforming approaches are known for accurately describing the surface motion. Furthermore, the mesh movement is enhanced by using a sophisticated mesh-update method that allows large mesh translation and augmenting the domain with new elements or, from the application point of view, extruding new material from the printed nozzle. The Cross-WLF viscosity model is used to represent the shear rate-dependent and the temperature-dependent behavior of the polymer material.