Keywords: homogenization, finite elements, singular systems, additive Schwarz preconditioning, coarse problem by aggregation.

This paper describes a numerical homogenization procedure which represents an efficient approach for capturing the overall characteristics of geo-composites and other materials. The implementation of the numerical homogenization procedure requires solving several non trivial tasks, such as the selection or development of an efficient iterative solution method which should be parallelizable, robust with respect to heterogeneity and jumps in coefficients and capable also of solving singular problems arising from the implementation of stress driven tests.

For parallelization, a two-level Schwarz type overlapping domain decomposition technique is used with the coarse problem constructed algebraically by aggregation and with inexact solution of the subproblems.

To be more specific, the motivation of the homogenization considered in the paper was an investigation of the properties of geocomposites, which arise from grouting the rock mass for the purpose of improving its mechanical behaviour. The investigation uses information about a complicated heterogeneous inner structure of a geo-composite that is provided by X-ray CT scan of samples. Based on this knowledge and the known properties of the components, numerical upscaling is implemented with the aid of the finite element method. The homogenized effective properties then represent the effect of grouting and enable an evaluation of the role of different factors.

From the computational point of view, the finite element analysis used for upscaling is very demanding, mainly from the necessity to solve problems on a very fine three-dimensional voxel mesh with millions of degrees of freedom and heterogeneous material composition with the use of different types of boundary conditions. The boundary conditions include pure Dirichlet ones for so called strain (displacement) driven tests and pure Neumann ones for stress (traction, force) driven tests.

In the case of linear material behaviour, the finite element analysis leads to the numerical solution of large symmetric finite element systems which are positive definite for Dirichlet and singular positive semidefinite for Neumann problems. We solve these systems using the conjugate gradient method preconditioned by the Schwarz domain decomposition technique. Then we investigate a suitable stabilization or regularization for solving singular symmetric positive semidefinite problems, construction of an auxiliary coarse problem by algebraic aggregation and the investigation of robustness of the two level additive Schwarz domain decomposition method when solving problems with heterogeneous material. We shall also use parallel computations and report parallel scalability of the method.

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