Multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous Galerkin formulation

When considering problems of dimensions close to the characteristic length of the material, the size effects can not be neglected and the classical (so-called first-order) multiscale computational homogenization scheme (FMCH) looses accuracy, motivating the use of a second-order multiscale computational homogenization (SMCH) scheme. This second-order scheme uses the classical continuum at the micro-scale while considering a second-order continuum at the macro-scale. Although the theoretical background of the second-order continuum is increasing, the implementation into a finite element code is not straightforward because of the lack of high-order continuity of the shape functions. In this work, we propose a SMCH scheme relying on the discontinuous Galerkin (DG) method at the macro-scale, which simplifies the implementation of the method. Indeed, the DG method is a generalization of weak formulations allowing for inter-element discontinuities either at the C0 level or at the C1 level, and it can thus be used to constrain weakly the C1 continuity at the macro-scale. The C0 continuity can be either weakly constrained by using the DG method or strongly constrained by using usual C0 displacement-based finite elements. Therefore, two formulations can be used at the macro-scale: (i) the full-discontinuous Galerkin formulation (FDG) with weak C0 and C1 continuity enforcements, and (ii) the enriched discontinuous Galerkin formulation (EDG) with high-order term enrichment into the conventional C0 finite element framework. The micro-problem is formulated in terms of standard equilibrium and periodic boundary conditions. A parallel implementation in three dimensions for non-linear finite deformation problems is developed, showing that the proposed method can be integrated into conventional finite element codes in a straightforward and efficient way.