Biomedical simulations often involve interfaces of geometrically complex shapes which are described by 3D image data. For this purpose, Composite Finite Elements (CFEs) for complicated domains, discontinuous coefficients, and their combination can be used. Their efficiency is due to the uniformly structured hexahedral grid given by the image data. The geometric complexity is treated by adapted FE basis functions without introducing additional degrees of freedom.
The treatment of boundary conditions is not straight-forward in this context because geometrically complicated boundaries are not resolved by the computational grid. We here describe how Dirichlet and Neumann boundary conditions can be incorporated in the CFE context. We moreover show that our approach numerically achieves the expected orders of convergence of the CFE approximation for increasing grid resolution, also in combination with discontinuous coefficients.
The methods explained here are used for two biomedical applications. First, radio-frequency ablation is an example where two scalar problems with complicated domains and discontinuous coefficients are considered. Second, a linearly elastic deformation of a vertebral disk is simulated in a two-scale model geometry also involving both a complicated domain and a discontinuous coefficient. The coefficient for the trabecular interior is obtained from numerical homogenization.