3D simulation of random materials

The Voronoi polyhedra model (3) is used to simulate polycrystalline media, which are later introduced into finite elements computation of deformation fields resulting from applied stresses (1, 2). This is an important step in the resolution by a computer of homogenization problems. To generate such microstructures, we propose an original method, with numerous extensions of this classical model Random polycrystals are made of a population of connected grains. To simulate this type of structures, it is necessary to use a random model, which accounts for the observed heterogeneity. Voronoi polyhedra model is convenient, for geometrical reasons (it presents fiat grain boundaries), and since it reproduces a germination and growth process, as required for the generation of polycrystals. Let E = {A(i)} be a set of random points P(x, y, z) corresponding to the centers of grains in a continuous domain D. The zone of influence of a point A(i) is defined by : iz(A(i)) = {P(x,y,z) is an element of D \ d(P, A(i)) < d(P, A(j)) For All j not equal i} where d(P-1, P-2) is the Euclidean distance between two points P-1 and P-2. By construction, this zone of influence builds the volume of the Voronoi polyhedron centered in A(i), and the set of zones of influence {iz(A(i))} builds a random tesselation of the domain D. A specific procedure was developed to build Voronoi polyhedra inside a discrete domain, made of a 3D voxel map. The polyhedra are generated ata given resolution, defined by the size of the three-dimensional domain. We have to affect to each voxel the number (or label) of the grain to which if belongs. In a first step, a germination process gives the locations (on the grid) of the centers of grains. Then, the Euclidean distance function of this set of points is calculated from an isotropic propagation starting from points, which is equivalent to a growth process (fig, 1a). Finally the segmentation of this image of distances enables us to accurately locate the grain boundaries namely their zones of influence (fig. 1c). For this step, the image of distances is considered as a topographical relief and the grain boundaries are obtained as the divide surfaces of the watersheds (a watershed being associated to each center). This algorithm is very efficient in its implementation based on hierarchical queues (9), which enables us to produce simulations in a short time. The proposed method can be easily extended. Periodic boundary conditions can be imposed, as in figures 1, 4, 5: the vicinity graph of the grid is simply made periodic before the calculations. Edge effects are suppressed by this process, and infinite media can be simulated. This type of periodic simulation is very useful for further finite element computations. Grain anisotropy can be generated from any deformation of the distance function (fig. 2). Thus are reproduced structures as obtained by a rolling process. If a phase, or a component, is randomly affected to each grain according to a given distribution, a multi-phase polycrystal is generated. The affectation can be made uniform over space, or can be made according to an underlying random medium. In the last case, the phases are affected non independently to the different polyhedra (fig. 3). Finally the distribution of grains can be made more regular : a minimal distance r between centers can be introduced; or, more generally a repulsion kernel (containing no other germ) can be given around every point of the process. In the last case. the location of centers is made sequentially, using an additional 3D image which records the forbidden locations In this way grains with a low size are eliminated and the distribution of the grain volumes is made much more uniform (fig. 5b).