Particle shape substantially influences the bulk behaviour of granular systems. Hence, much scientific interest has been devoted to the adoption of non-spherical fundamental particles in discrete element method simulations. Two examples of such particles are polyhedra, which are highly angular, and superquadrics, which are best suited to simulate rounded particles. It is desirable to use both types of particle together in a simulation to capture the broadest possible range of particle shapes. In this paper, a novel contact detection algorithm is presented for a convex polyhedron and superquadric. This algorithm was implemented in a C++ code which was used to verify the correctness of the algorithm and evaluate its efficiency using the Monte Carlo method. The proposed contact detection algorithm is particularly efficient for many-faceted polyhedra as the effect of increasing the number of faces on the evaluation time is small. (C) 2019 Elsevier B.V. All rights reserved.