Numerical methods such as the finite element method (FEM) are commonly used for mechanical problems, discretizing the geometry with a boundary-conforming mesh. However, an effficient simulation of highly irregular and heterogeneous microstructures such as cemented granular materials (CGM) is a challenging task. The FEM would require to a very fine mesh – especially at the material interfaces – in order to resolve the geometry, leading to large linear system and high computational costs. Alternatively, immersed methods like the finite cell method (FCM) have become attractive during the recent years. The FCM combines high-order hierarchical shape functions with the fictitious domain approach, meshing complex geometries with a Cartesian grid. For CGM, which are heterogeneous and described by 3D images, acquired from X-ray computer tomography (CT), several issues arise. Firstly, the staircase geometry description of the 3D images leads to inaccurate results, such as artificial singularities in the stresses. This issue is resolved by global L2-projection, which approximates the non-smooth geometry by a level-set function, resulting in a smooth geometry reconstruction. Furthermore, an extended L2-projection approach for heterogeneous materials is presented. Secondly, heterogeneous materials lead to weak discontinuities at the material interfaces, i.e. kinks in the displacements and jumps in the strains and stresses occur. Since the FCM uses piecewise smooth polynomials, it is unable to capture the discontinuous solution and therefore, the convergence rate deteriorates. To tackle this issue, the FCM is extended by local enrichment. This approach helps to capture the jumps of the stresses and thus, simulate heterogeneous materials reliably. In this contribution, the extended L2-projection and the local enrichment are combined in the context of the FCM, with the goal of accurately simulating heterogeneous image-based materials. The proposed approaches are tested on benchmarks first and are finally applied to CGM.