Nguyen Lam, Stoter Stein, Baum Thomas, Kirschke Jan, Ruess Martin, Yosibash Zohar, Schillinger Dominik
Department of Civil, Environmental, and Geo- Engineering, University of Minnesota, Minneapolis, MN 55455, USA.
Department of Neuroradiology, Klinikum rechts der Isar, Technische Universität München, Munich, Germany.
Int J Numer Method Biomed Eng. 2017 Dec;33(12). doi: 10.1002/cnm.2880. Epub 2017 May 11.
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.
体素有限元法使用非拟合有限元网格和体素求积规则，将计算机断层扫描数据无缝转换为患者特异性的骨骼离散模型。然而，该方法仍需要对边界表面进行显式参数化，以施加牵引力和位移边界条件，这构成了自动化的潜在障碍。我们探索了一种基于相场的公式，用于在扩散意义上施加牵引力和位移约束。其核心组件是一个由艾伦-卡恩问题的亚稳相场解生成的扩散几何模型，该模型将成像数据作为初始条件。然后，利用边界及其梯度的相场近似，将变分公式中的所有边界项转换为体积项。我们表明，在体素有限元法的背景下，如果固有长度尺度，即相场的界面宽度、体素间距和网格尺寸，适当地相关，则扩散边界条件与在显式尖锐表面上定义的边界条件具有相同的精度。我们通过分析人体股骨和椎体的应力来证明新方法的灵活性。
