Keywords: fiber shape optimization, composites, special formulation, parallel computing.

In this study shape optimization of fibers in a periodic composite structure is presented. The optimization is based on the minimization of material properties of phases in the composite and the overall (effective) properties, which are given a priori. Such a problem is important in a wide range of applications, including biomedicine and biophysics. The classical problem of localization and homogenization is improved in such a manner that the computation of the optimal shape is simpler. Since the response of unit movements of the fiber boundary are calculated, the process of optimization can be distributed and offers a general idea of the optimization of more complex problems.

In this paper a new procedure for homogenization of composites is proposed. Contrary to Reference [1], where a beam structure is considered for the optimization of minimum displacements and the highest bearing capacity, the contrary case is taken into account. The overall properties are known a priori, while either no restriction on the shape is put forward or some additional constraints are formulated according to the sense of the problem. Based on the finite element method and its improvement suiting for the solution of the stated problem some formulations are suggested and after that the numerical solution is outlined. Special properties for the distribution of strains and stresses (or concentration factors) on a unit cell are used. An averaging process including integration over the unit cell, particularly in the case of fiber reinforced concrete (when the fiber ratio is relatively small) follows the fact that the stresses "relax" on the matrix and their distribution converges to a constant except for a zone in the vicinity of the fiber-matrix interfacial surface. These properties of the stress and strain distributions are fully utilized in this paper.

A similar idea published by Prochazka [2], is then used, in this case using the finite element method instead of the boundary element method. In some cases (in view of the solvability) additional constraints have to be involved: the boundaries of the unit cell and of the fiber must remain disjoint.

1

P.P. Prochazka, V. Dolezel, T.S. Lok, "Optimal shape design for minimum Lagrangian", Engineering Analysis with Boundary Elements, 33, 447-455, 2009. doi:10.1016/j.enganabound.2008.09.003

2

P. Procházka, "Homogenization of linear and of debonding composites using the BEM", Engineering Analysis with Boundary Elements, 25(9), 753-769, 2001. doi:10.1016/S0955-7997(01)00066-2

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