The positions on the grid leading to overlap inclusion positions by allowing selection only among the points This allows the addition of a new inclusion at each linked to any probability relative to the actual density Additional neighboring distance conditions can be enforced u ++ 2: where ++ 2:and ++ ;are the minimum and maximum distance imposed of the added inclusion (Fig. 2a). To optimized by minimizing the distances and its third nearest neighbors in 3D. are used for this purpose with the same type of condition (a) Figure 2. (a) Non-overlap and first neighbor distance criteria to restrict the random position of the new inclusion to be added (the function LS neighbor distance criteria to optimize spatial organization to obtain dense packings (the function 9, 3.2. Inclusion morphing Marginal corrections required to adjust the inclusion volume fraction or shapes and more substantial modifications allowing to produce polycrystalline or cellular microstructures motivate the development of a tool enabling the morphing of inclusions once t generated by sequential addition. At this stage, the inclusions neighborhood is completely determined and can be used to modify their shape according to inter complete expansion of inclusions until va polycrystal-like microstructures. The morphing technique is strongly based on level set functions built during the sequential addition can be contoured to extract updated inclusions to form a polycrystal morphology is used here to function < vanishes at points of equal distance between two nearest inclusion zero level set of this function thus determines a Voronoï inclusion and points closer to it than to other 13th International Conference on Fracture June 16 -5- on the grid leading to overlap with existing inclusions can be excluded for the random s by allowing selection only among the points satisfying the condition 9, : => a new inclusion at each trial, and the generation probability relative to the actual density, but rather to the number of added inclusion Additional neighboring distance conditions can be enforced using 9, such as > ? 9, : ? ++ ; > the minimum and maximum distance imposed from ). To increase packing density, the spatial organization has t optimized by minimizing the distances of the added inclusion to its second nearest neighbor neighbors in 3D. The corresponding distance functions 9, are used for this purpose with the same type of condition (Fig. 2b). (b) overlap and first neighbor distance criteria to restrict the random position of the (the function LS1(x) is represented in the insert), (b) optimize spatial organization to obtain dense packings (the function @ : is represented in the insert) development of a tool enabling the morphing of inclusions once their population is entirely determined and can be used to modify their shape according to inter-inclusion distance rules. complete expansion of inclusions until vanishing the inter-grain joint thickness like microstructures. The morphing technique is strongly based on level set functions. The 9, during the sequential addition process are used to construct a function updated shapes of the inclusions. The case of complete expansion of inclusions to form a polycrystal morphology is used here to illustrate the methodology. The < : 9, : − 9,@ : vanishes at points of equal distance between two nearest inclusions and is negative elsewhere. The zero level set of this function thus determines a Voronoï-like diagram, each cell enclosing an inclusion and points closer to it than to other inclusions. If the initial inclusion distribution is a June 16–21, 2013, Beijing, China can be excluded for the random satisfying the condition (Fig.2a) (14) the generation cost is therefore not the number of added inclusions. such as for instance (15) from the first neighbor , the spatial organization has to be to its second nearest neighbor in 2D @ : and 9,A : First and second heir population is entirely inclusion distance rules. A grain joint thickness allows forming : and 9,@ : uct a function < : that inclusions. The case of complete expansion of the methodology. The (16) and is negative elsewhere. The like diagram, each cell enclosing an . If the initial inclusion distribution is a
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