13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- to them. At equilibrium, the discretized system of equations for the transport problem can be condensed at the control nodes . / 0 1234 5 ( 6# 6) 5 5 (11) where 01234 5 results from the condensation of the entire RVE stiffness values. Using periodicity, the averaged flux is obtained from the 'reaction' fluxes at the controlling nodes as .'7 1 * . / 2 , - 1 . / 2 (12) Upon substitution of the two previous relationships, one identifies the (averaged) permeability as '78 2 0 1234 5 6 5 5 5 (13) As in the mechanical case, any modelling choice (constitutive laws, discretisation technique) can be adopted for the fine-scale modelling. 3. Representative volume element generation The RVE generator tool is mainly designed based on distance fields and level set functions as presented in [3] for 2D RVEs, and extended in [4] for 3D simulations. A random distribution of inclusions is first generated, which is subsequently morphed to produce a grain-like structure. 3.1. Inclusion packing The inclusion packing is the first step of the RVE generation method that gives the basis for the microstructural spatial arrangement. It allows incorporating prescribed volume fractions and/or grain size distributions as input parameters. The geometry used for the shape of inclusions is arbitrary and can be randomly generated through a parameterization, or explicitly defined from existing data (e.g. in order to use data from Computed X-Ray Tomography). The problem of filling a container with a given volume fraction of inclusions while respecting prescribed size distributions and grain shapes is achieved here using a sequential addition of inclusions, improved by the use of distance fields. In the classical RSA algorithm, each loop generates randomly a trial position in the RVE for the next inclusion to be added. This inclusion is then verified to ensure no interpenetration with previously added inclusions when placed at the trial position, in which case it is rejected and another trial position is generated. Costly computational operations (overlap and distance evaluation) are required at each trial, but few trials lead to a successful inclusion addition, especially when dense packing must be reached. This original sequential addition methodology can be dramatically improved using distance fields. Instead of a purely random trial position, a set of discrete positions satisfying a priori the non-overlapping and neighboring distance conditions is used to select new inclusion locations. This set is built using the nearest neighbor distance function 9, (see Fig. 2a) which is maintained on a structured grid of points : at each inclusion addition. The radius r of the smallest enclosing circle (or sphere) of the new inclusion is used as an indicator of its size.
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