Methods and systems for providing accurate, scalable, and predictive 3D printing simulations using numerical methods for part-level simulations. Complex parts can be discretized into finite elements using independent and arbitrary meshing. The real additive manufacturing tooling path and printing time of a printing machine are simulated and applied to the mesh of finite elements using an intersection module that combines the finite element mesh with the tool path information of the printing machine in a geometric sense. This allows for localized heating effects to be simulated very accurately, and for cooling assessments to be precisely computed given the intersection module's computation of partial facets and volumes of the finite elements at any given time in the printing simulation.