AbstractComputational homogenization is the gold standard for concurrent multi-scale simulations (e.g., FE2) in scale-bridging applications. Often the simulations are based on experimental and synthetic material microstructures represented by high-resolution 3D image data. The computational complexity of simulations operating on such voxel data is distinct. The inability of voxelized 3D geometries to capture smooth material interfaces accurately, along with the necessity for complexity reduction, has motivated a special local coarse-graining technique called composite voxels (Kabel et al. Comput Methods Appl Mech Eng 294: 168–188, 2015). They condense multiple fine-scale voxels into a single voxel, whose constitutive model is derived from the laminate theory. Our contribution generalizes composite voxels towards composite boxels (ComBo) that are non-equiaxed, a feature that can pay off for materials with a preferred direction such as pseudo-uni-directional fiber composites. A novel image-based normal detection algorithm is devised which (i) allows for boxels in the firsts place and (ii) reduces the error in the phase-averaged stresses by around 30% against the orientation cf. Kabel et al. (Comput Methods Appl Mech Eng 294: 168–188, 2015) even for equiaxed voxels. Further, the use of ComBo for finite strain simulations is studied in detail. An efficient and robust implementation is proposed, featuring an essential selective back-projection algorithm preventing physically inadmissible states. Various examples show the efficiency of ComBo against the original proposal by Kabel et al. (Comput Methods Appl Mech Eng 294: 168–188, 2015) and the proposed algorithmic enhancements for nonlinear mechanical problems. The general usability is emphasized by examining various Fast Fourier Transform (FFT) based solvers, including a detailed description of the Doubly-Fine Material Grid (DFMG) for finite strains. All of the studied schemes benefit from the ComBo discretization.