The process of drawing an optical fiber from a polymer preform is still not completely understood,although it represents one of the most critical steps in the process chain for the fabrication of microstructuredpolymer optical fibers (mPOFs). Here we present a new approach for the numerical modelling of the fiber drawingprocess using a fully three-dimensional and time-dependent finite element method, giving significant insightinto this widely spread mPOF production technique. Our computational predictions are physically based on theviscoelastic fluid dynamics of polymers. Until now the numerical modelling of mPOF drawing has mainly beenbased on principles, such as generalized Newtonian fluid dynamics, which are not able to cope with the elasticcomponent in polymer flow. In the present work, we employ the K-BKZ constitutive equation, a non-linearsingle-integral model that combines both elastic and viscous ideas and can appropriately describe the physics ofpolymers under processing.