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This work presents a two-scale homogenization procedure to analyze three dimension composite structures by the finite element method. The theory implemented is compared with other micro-structural formulations: micro models and the serial-parallel mixing theory, in terms of result accuracy and computational cost. The comparison shows that for linear analysis, the homogenization proposed is an excellent alternative to the other formulations considered. Its computational cost is substantially lower than the one required by the micro-model and it is able to capture some micro-structural phenomena that it is not automatically recorded by the serial-parallel mixing theory. It will also be shown that the extension of the proposed theory to the non-linear range stills represents a challenge. The major limitation is its prohibitive computational cost because it requires solve the sub scale at each gauss point and each load step. However the comparison shows that this cost is in terms of CPU time but not in terms of memory. Based on the results obtained, it can be concluded that the homogenization method is an excellent alternative for the simulation of materials with complex micro structures. The method is also very promising for non linear simulations, * Corresponding author

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The use of multi-scale procedures is encouraged by the continuous increase of computational capacity, but it is still a challenge performing a non-linear analysis of a real composite structure without the aid of large computers. This work proposes a strategy to conduct non-linear two-scale analysis in an efficient way. The proposed method considers that in a large structure, in general, material non-linear processes only take place in a localized region (or in a reduced number of finite elements, if a FE method is used). The strategy determines the elements that require a non-linear analysis defining of a non-linear activation function that accounts for the failure of the most critical point in the microstructure. The procedure conserves the dissipated energy through the scales, being mesh independent as the mesh objectivity concept is extended to the microstructure. The validity of the strategy proposed is proved with the analysis of academic examples showing not only the mesh independency but also the reduction of computational cost. Finally, an industrial composite component is solved using a standard computer, showing that the proposed strategy is capable of reducing the computational cost from 32 days and 14 hours (required by a classical multi-scale method) to less than 12 hours.