An additive manufacturing technique, known as Direct Ink Write (DIW) allows the creation of complex geometries for many technologies by depositing an “ink” material through a syringe onto a substrate for a chosen pattern. The inks used in DIW often have complicated rheological properties and thus presents many challenges for computational models. These challenges include the highly nonlinear behavior of shear-thinning materials, as well as tracking of the complex material interfaces as the ink is deposited onto the substrate. Here, we will simulate the DIW process using the Galerkin finite-element method (GFEM) to solve the transient, three-dimensional Navier-Stokes equation using a non-Newtonian viscosity model to capture the effects of the shear-thinning ink material. To track the ink/air/substrate interfaces we use Conformal Transient h-r Unstructured Adaptive Mesh Refinements (cThruAMR) method, which is new method for tracking material interfaces while maintaining high-quality elements connected to the interface surfaces. cTHruAMR is a conformal meshing algorithm that creates sharp boundaries on the material interfaces, which eases application of boundary conditions such a capillarity or temperature jumps between phases. In this talk, we will present the results of both two- and three-dimensional simulations of the DIW process, observing various flow features and other physical phenomena for several printing patterns and ink formulations for both Newtonian and non-Newtonian viscosity models.