Calculating contact forces between complex shapes for performing Discrete Element Method (DEM) simulations is a long standing problem with no unique ideal solution. In this work, a new method to calculate interactions between arbitrary rounded bodies is presented. Each body is represented by a triangulated surface mesh, in which each triangle is associated with a unique radius of curvature. Then, normal contact forces are calculated by numerically integrating a (Hertz) contact pressure formulation over the contact area between two contacting particles. This results in a mechanistic contact description that is converging with refinement of a given triangulation and directly uses physical material properties as parameters of the contact model. After showing convergence upon mesh refinement towards the Hertzian solution, the error for non-spherical curvatures is investigated and the new model is compared with an indentation experiment of a pear-shaped object. Finally, the method is demonstrated in a simulation of gravitational packing by simulating packing of spheres, pear-shaped as well as gummy bear-shaped objects.