Phase-field boundary conditions for the voxel finite cell method: surface-free stress analysis of CT-based bone structures

摘要

The voxel finite cell method employs unfitted finite element meshes and voxel quadrature rules to seamlessly\udtransfer CT data into patient-specific bone discretizations. The method, however, still requires the explicit\udparametrization of boundary surfaces to impose traction and displacement boundary conditions, which\udconstitutes a potential roadblock to automation. We explore a phase-field based formulation for imposing\udtraction and displacement constraints in a diffuse sense. Its essential component is a diffuse geometry model\udgenerated from metastable phase-field solutions of the Allen-Cahn problem that assumes the imaging data as\udinitial condition. Phase-field approximations of the boundary and its gradient are then employed to transfer\udall boundary terms in the variational formulation into volumetric terms. We show that in the context of the\udvoxel finite cell method, diffuse boundary conditions achieve the same accuracy as boundary conditions\uddefined over explicit sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field,\udthe voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method\udby analyzing stresses in a human femur and a vertebral body.