PSI - Issue 6

Anna Kudimova et al. / Procedia Structural Integrity 6 (2017) 301–308 Author name / Structural Integrity Procedia 00 (2017) 000–000

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B e e ∗ φ are the matrices and the row vectors of approximate shape functions, respectively, defined on separate finite elements. The vectors F u and F φ in (16), (17) are obtained from the boundary conditions, the corresponding right parts of the weak statements, and the finite element approximations. Note that in ACELAN we use an e ff ective algorithm with symmetric quasi-definite matrices [Belokon et al. (2000, 2002); Nasedkin (2010, 2015)] for solving finite element Eqs. (16), (17) [Benzi et al. (2005); Benzi and Wathen (2008); Vanderbei (1995)]. All the procedures that we need in finite element manipulations (the degree of freedom rotations, mechanical and electric boundary condition settings, etc.) can be provided in symmetric form. u = L ( ∇ ) · N e ∗ u , B e φ = ∇ N e ∗ φ ; N e ∗ u , N

4. Modeling of Representative Volumes

In ACELAN-COMPOS we simulate mixed composite materials using cubic eight-node or twenty-node elements (octants) in the representative volume. Initial volume in the form of a large-scale cube is divided into octants using Octree algorithm, where the size and the number of the elements may vary depending on the physical properties of the material or the needed percentage of pores or inclusions. The octants can easily be used as finite elements on regular mesh. The structural elements of the same material are united into the composite component. Each component has a connectivity property: any element belonging to the component is reachable from any other component element from the transition between the adjacent elements. This approach allows us to model two-phase composite materials with 3–3 connectivity. Examples of representative volumes for these composites are shown in Fig. 1 for di ff erent number n of elements along the axis, and for p = 20% of volume fraction of second phase material.

Fig. 1. Representative volumes of 3–3 composites: full volume, n = 8 (a); second phase, n = 8 (b); full volume, n = 32 (c); second phase, n = 32 (d).

Fig. 2. Representative volumes of 3–0 composites: full volume, n = 16 (a); second phase, n = 16 (b); full volume, n = 32 (c); second phase, n = 32 (d).

The second algorithm is designed for the case when the second phase of the composite (inclusion or pore) is dis tributed in the form of isolated areas in the shape of granules. The maximal and minimal sizes and overall percentage of the granules can be set as input parameters. At each step of the iteration process, we select random key point in the cluster, with a restriction that forbids merging of granules. Using this restriction, we select candidates for the next