Discrete element Newtonian dynamics simulations have been carried out of filling and discharge under gravity of non-cohesive discs (in two dimensions) and spheres (in three dimensions) from model hoppers. The current model improves that developed previously by us (Langston et al., 1994) in several respects. We introduce a continuous and gradual hopper filling method, a more realistic normal-tangential interaction between the particles, particle size polydispersity, and the model is extended from two to three dimensions (3D). The hopper discharge rate has been computed as a function of material head height, outlet size and the hopper half-angle. The model results are, in general, in very good agreement with established literature empirical predictions. The hopper wall stresses have been compared in the static state after filling and in the dynamic state during discharge. Generally there is encouraging agreement with predictions from the continuum differential slice force balance method, with significant improvements over our previous work. We have also observed, for the first time in a discrete element simulation of hoppers, the appearance of rupture zones within the material and associated wall stress peaks where the rupture zones intersect with the hopper wall. We consider that the current model is more successful than the previous one because the particle interactions include a much greater level of frictional "engagement" at low loads, with less variation at high loads.