Large-scale numerical simulation using the Discrete Element Method (DEM) contributes to improve our understanding of granular flow dynamics involved in many industrial processes and geophysical flows. In industry, it leads to an enhanced design and an overall opti-misation of the corresponding equipment and process. Most of DEM simulations in the literature have been performed using spherical particles. A limited number of studies dealt with non-spherical particles, even less with non-convex particles. Even convex bodies do not always represent the real shape of certain particles. In fact, more complex shaped particles are found in many industrial applications as, e.g., catalytic pellets in chemical reactors or crushed glass debris in recycling processes. In Grains3D-Part I [39], we addressed the problem of convex shape in granular simulations while in Grains3D-Part II [33], we suggested a simple through efficient parallel strategy to compute systems with up to a few hundreds of millions of particles. The aim of the present study is to extend even further the modeling capabilities of Grains3D towards non-convex shapes, as a tool to examine the flow dynamics of granular media made of non-convex particles. Our strategy is based on decomposing a non-convex shaped particle into a set of convex bodies, called elementary components. We call our method glued or clumped convex method, as an extension of the popular glued spheres method. Essentially, a non-convex particle is constructed as a cluster of convex particles, called elementary components. At the level of these elementary components of a glued convex particle, we employ the same contact detection strategy based on a Gilbert-Johnson-Keerthi algorithm and a linked-cell spatial sorting that accelerates the resolution of the contact, that we introduced in [39]. Our glued convex model is implemented as a new module of our code Grains3D and is therefore automatically fully parallel. We illustrate the new modeling capabilities of Grains3D in two test cases: (i) the filling of a container and (ii) the flow dynamics in a rotating drum.