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

Abstract

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.

The diffuse Nitsche method: Dirichlet constraints on phase‐field boundaries

“…The curves plotted in Figure A and B also demonstrate that the maximum accuracy directly correlates with the length‐scale parameter ε used in the diffuse phase‐field representation . We note that the same characteristic convergence behavior has been demonstrated for diffuse Neumann boundary conditions by Nguyen and collaborators in Nguyen et al…”

“…Therefore, we can summarize a desirable relation between the 3 inherent length scales as follows The constraint on the mesh size h by the voxel spacing Δ in that automatically follows from the above considerations is in line with the limitation that the accuracy of finite element schemes based on voxel quadrature cannot be increased by mesh refinement beyond the voxel resolution. ()…”

“…For Neumann‐type boundary and interface conditions, this strategy leads to a straightforward phase‐field approximation. () For Dirichlet‐type constraints, most of the attention has been focused on penalty‐type approaches. ()…”

“…In this context, we combine the diffuse Nitsche method with voxel quadrature strategies to evaluate volume integrals directly on imaging data. () We derive suitable relations between the 3 different length scales involved, that is, the length scale of the phase‐field that controls the width of the diffuse boundary region, the voxel spacing that represents the resolution of the imaging data, and the mesh size that controls the accuracy of the approximation of the physics‐based solution fields. Finally, we illustrate the versatility of this approach by a virtual compression test of a patient‐specific bone structure that requires the imposition of Dirichlet boundary conditions on a complex CT‐based geometry …”

A diffuse modeling approach for embedded interfaces in linear elasticity

“…Instead of a simple interpolation, a homogenization is applied to compute the effective material parameters in the diffuse interface region, similar to phase field models. The resulting method can be interpreted as a diffuse interface method [33][34][35][36][37] with an enhanced treatment of the interface mechanics.…”