Multiscale Computational Homogenization: Review and Proposal of a New Enhanced-First-Order Method

The continuous increase of computational capacity has encouraged the extensive use of multiscale techniques to simulate the material behaviour on several fields of knowledge. In solid mechanics, the multiscale approaches which consider the macro-scale deformation gradient to obtain the homogenized material behaviour from the micro-scale are called first-order computational homogenization. Following this idea, the second-order FE2 methods incorporate high-order gradients to improve the simulation accuracy. However, to capture the full advantages of these high-order framework the classical boundary value problem (BVP) at the macro-scale must be upgraded to high-order level, which complicates their numerical solution. With the purpose of obtaining the best of both methods i.e. first-order and second-order, in this work an enhanced-first-order computational homogenization is presented. The proposed approach preserves a classical BVP at the macro-scale level but taking into account the high-order gradient of the macro-scale in the micro-scale solution. The developed numerical examples show how the proposed method obtains the expected stress distribution at the micro-scale for states of structural bending loads. Nevertheless, the macro-scale results achieved are the same than the ones obtained with a first-order framework because both approaches share the same macro-scale BVP.