For the numerical simulation in continuum mechanics the Composite Finite Element (CFE) method allows an effective treatment of problems where material parameters are discontinuous across geometrically complicated interfaces. Instead of complicated and computationally expensive tetrahedral meshing, specialized CFE basis functions are constructed on a uniform hexahedral grid. This is a convenient approach in practice because frequently in biomechanics geometric interfaces are described via 3D image data given as voxel data on a regular grid. Then, for a particular coupling condition that depends on an underlying physical conservation law and the local geometry of the interface, one constructs CFE basis functions that are capable of representing functions satisfying this coupling condition. In this paper we present in detail this construction for heat conduction and linear elasticity as scalar and vector-valued model problems. Furthermore, we show first numerical results.