Granular flow is widely found in nature or industrial production. Although the external driving force significantly affects the dynamic behavior of a granular system, a large number of numerical simulations have been conducted to study granular flows driven by gravity. In this study, a superquadric equation was used to construct spherical and cylindrical elements, and the flow processes of granular materials under external pressure were simulated by the discrete element method. To examine the validity of the DEM model, the Janssen effect of spherical particles, the static packing of cylindrical particles and the flow process of spherical particles under external pressure are simulated and compared with the previous experimental and theoretical results. Subsequently, the effects of blockiness, orifice diameter, and particle friction on the flow characteristics are investigated. Results show that the flow rate of spherical particles increases as the external pressure and opening diameter increase or the particle friction decreases. However, the flow rate of cylindrical particles decreases as the blockiness parameter increases, and the external pressure has little effect on the flow rate of the cylindrical particles when the blockiness parameter is greater than 4. Furthermore, the external pressure causes a change in the flow pattern of granular systems. In a gravity-driven granular flow, cylindrical particles appear in funnel flow, and spherical particles in both mass and funnel flows. In a pressure-driven granular flow, spherical particles appear in mass flow, and cylindrical particles in both mass and funnel flows. The critical height of the transition between mass and funnel flows decreases with increasing external pressure and eventually reaches a steady state. Meanwhile, the critical height increases with the blockiness parameter, which indicates that more cylindrical than spherical particles appear in funnel flow. Finally, the basic flow characteristics of granular materials under external pressure are further analyzed by the velocity uniformity index, the normal contact force between particles, and the bottom pressure. Overall, the numerical results are useful for understanding the changes in the flow characteristics of spherical and cylindrical granular materials under external pressure, and further provide guidance for the appropriate design and optimization of cylindrical hoppers.