Numerical simulation based on mathematical models is an important pillar for enhancing the understanding of permeation processes in the skin. To adequately resolve the complex geometrical structure of the skin, special models based on tetrakaidecahedral cells have been suggested. While these models preserve many of the desirable properties of the underlying geometry, they impose challenges regarding mesh generation and solver robustness. To improve robustness of the used multigrid solver, we propose a new mesh and hierarchy structure with good aspect ratios and angle conditions. Furthermore, we show how those meshes can be used in scalable massively parallel multigrid based computations of permeation processes in the skin.