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 simula te this type of structures, it is necessary to use a random model, which ac counts for the observed heterogeneity. Voronoi polyhedra model is convenien t, for geometrical reasons (it presents fiat grain boundaries), and since i t reproduces a germination and growth process, as required for the generati on of polycrystals. Let E = {A(i)} be a set of random points P(x, y, z) corresponding to the ce nters 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 p olyhedron centered in A(i), and the set of zones of influence {iz(A(i))} bu ilds a random tesselation of the domain D. A specific procedure was developed to build Voronoi polyhedra inside a disc rete 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 belong s. 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 se t of points is calculated from an isotropic propagation starting from point s, which is equivalent to a growth process (fig, 1a). Finally the segmentat ion of this image of distances enables us to accurately locate the grain bo undaries namely their zones of influence (fig. 1c). For this step, the imag e of distances is considered as a topographical relief and the grain bounda ries are obtained as the divide surfaces of the watersheds (a watershed bei ng associated to each center). This algorithm is very efficient in its impl ementation based on hierarchical queues (9), which enables us to produce si mulations in a short time. The proposed method can be easily extended. Periodic boundary conditions ca n be imposed, as in figures 1, 4, 5: the vicinity graph of the grid is simp ly made periodic before the calculations. Edge effects are suppressed by th is process, and infinite media can be simulated. This type of periodic simu lation is very useful for further finite element computations. Grain anisot ropy can be generated from any deformation of the distance function (fig. 2 ). Thus are reproduced structures as obtained by a rolling process. If a ph ase, or a component, is randomly affected to each grain according to a give n distribution, a multi-phase polycrystal is generated. The affectation can be made uniform over space, or can be made according to an underlying rand om medium. In the last case, the phases are affected non independently to t he 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 gi ven around every point of the process. In the last case. the location of ce nters is made sequentially, using an additional 3D image which records the forbidden locations In this way grains with a low size are eliminated and t he distribution of the grain volumes is made much more uniform (fig. 5b).