Numerical Homogenization of Heterogeneous Anisotropic Linear Elastic Materials

The numerical homogenization of anisotropic linear elastic materials with strongly heterogeneous microstructure is studied. The developed algorithm is applied to the case of trabecular bone tissue. In our previous work [1], the orthotropic case was considered. The homogenized anisotropic tensor is transformed according to the principle directions of anisotropy (PDA). This provides opportunities for better interpretation of the results as well as for classification of the material properties. The upscaling procedure is described in terms of six auxiliary elastic problems for the reference volume element (RVE). Rotated trilinear Rannacher-Turek finite elements are used for discretization of the involved subproblems. A parallel PCG method is implemented for efficient solution of the arising large-scale systems with sparse, symmetric, and positive semidefinite matrices. Then, the bulk modulus tensor is computed from the upscaled stiffness tensor and its eigenvectors are used to define the transformation matrix. The stiffness tensor of the material is transformed with respect to the PDA which gives a canonical (unique) representation of the material properties. Numerical experiments for two different RVEs from the trabecular part of human bones are presented.