Geometry-dependent reduced-order models for the computation of homogenized transfer properties in porous media

This article investigates reduced-order models (ROMs) based on proper orthogonal decomposition for efficient computation of the homogenized diffusion properties of saturated porous media. The homogenized tensor, whose classical expression may be obtained from periodic homogenization techniques, is computed by solving a local problem on an elementary cell that includes one or several circular solid inclusions. The cost of the repeated resolution of this problem by the finite element method for different inclusions radii may be important, and classical model order reduction methods based on the computation of a spatial basis cannot be applied directly. The method proposed in this work to cope with the variability of circular inclusions relies on the introduction of a transformation from a reference domain to the physical domain that admits an exact affine decomposition in the three-dimensional cases, allowing to split the problem into an offline learning phase and an online evaluation phase which does not depend on the number of degrees of freedom of the original full-order solution. Approximate affine decomposition for two-dimensional cases is also provided with an explicit estimation of the truncation error. The efficiency of the proposed algorithm in terms of accuracy and of computing time is evaluated first for 2D and 3D isotropic and anisotropic elementary cells with a single inclusion, and second for a 2D anisotropic cell with multiple inclusions. Furthermore the ROM is used to estimate in quasi-real time the homogenized diffusion tensor for a given probability distribution of the geometry parameters.