Parallel MIC(0) Preconditioning for Numerical Upscaling of Anisotropic Linear Elastic Materials

Numerical homogenization is used for upscaling of the linear elasticity tensor of strongly heterogeneous microstructures. The implemented 3D algorithm 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. The implemented preconditioner is based on modified incomplete Cholesky factorization MIC(0).

The numerical homogenization scheme is derived on the assumption of periodic microstructure. This implies periodic boundary conditions (PBCs) on the RVE. From algorithmic point of view, an important part of this study concerns the incorporation of PBCs in the parallel MIC(0) solver.

Numerical upscaling results are shown. The test problem represents a trabecular bone tissue, taking into account the elastic response of the solid phase. The voxel microstructure of the bone is extracted from a high resolution computer tomography image. The presented results evidently demonstrate that the bone tissues could be substantially anisotropic.

The computations are performed on IBM Blue Gene/P machine at the Bulgarian Supercomputing Center.