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

Lam Nguyen, Stein Stoter, Thomas Baum, Jan Kirschke, Martin Ruess, Zohar Yosibash, Dominik Schillinger*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

37 Scopus citations


The voxel finite cell method uses unfitted finite element meshes and voxel quadrature rules to seamlessly transfer computed tomography data into patient-specific bone discretizations. The method, however, still requires the explicit parametrization of boundary surfaces to impose traction and displacement boundary conditions, which constitutes a potential roadblock to automation. We explore a phase-field–based formulation for imposing traction and displacement constraints in a diffuse sense. Its essential component is a diffuse geometry model generated from metastable phase-field solutions of the Allen-Cahn problem that assumes the imaging data as initial condition. Phase-field approximations of the boundary and its gradient are then used to transfer all boundary terms in the variational formulation into volumetric terms. We show that in the context of the voxel finite cell method, diffuse boundary conditions achieve the same accuracy as boundary conditions defined over explicit sharp surfaces, if the inherent length scales, ie, the interface width of the phase field, the voxel spacing, and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human femur and a vertebral body.

Original languageEnglish
Article numbere2880
JournalInternational Journal for Numerical Methods in Biomedical Engineering
Issue number12
StatePublished - Dec 2017
Externally publishedYes


FundersFunder number
National Science Foundation1565997, CISE-1565997, 1651577
University of Minnesota
Horizon 2020 Framework Programme637164
Minnesota Supercomputing Institute, University of Minnesota
European Research Council
Deutsche ForschungsgemeinschaftBA 4085 2/1
Horizon 2020


    • diffuse boundary methods
    • femur
    • patient-specific simulation
    • phase-fields
    • vertebra
    • voxel finite cell method


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