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

Abstract

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 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 C 0 level or at the C 1 level, and it can thus be used to constrain weakly the C 1 continuity at the macro-scale. The C 0 continuity can be either weakly constrained by using the DG method or strongly constrained by using usual C 0 displacement-based finite elements. Therefore, two formulations can be used at the macro-scale: (i) the full-discontinuous Galerkin formulation (FDG) with weak C 0 and C 1 continuity enforcements, and (ii) the enriched discontinuous Galerkin formulation (EDG) with high-order term enrichment into the conventional C 0 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.

On the algorithmic and implementational aspects of a Discontinuous Galerkin method at finite strains

“…We highlight three key features of the stabilization terms as derived in [1]: (6) induce a weighted average definition of the numerical flux in (4)- (5). Through the explicit dependence on the sector ω (α)…”

“…However, theorems and analyses conducted in the linear context do not always carry over to the nonlinear context. For example, many existing DG methods for finite strains, including those for hyperelasticity [3], plasticity [4], and second-order computational homogenization [5], possess a nonsymmetric incremental weak form. Loss of symmetry has been shown in the linear context [6] to upset adjoint consistency as well as yielding suboptimal L 2 error convergence rates.…”

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

“…This assumption is particularly valid when macrogradients remain small and material failure does not occur. The second-order computational homogenization partly alleviates the assumption of scale separation by taking the gradient of the macrodeformation gradient tensor into account [246][247][248][249][250]. Furthermore, second-order computational homogenization introduces a physical length to the microscale that is missing in the first-order homogenization.…”

Unified treatment of microscopic boundary conditions and efficient algorithms for estimating tangent operators of the homogenized behavior in the computational homogenization method

“…In the nonlinear range, the system of equations (75) is iteratively resolved by the multiplier elimination method [57,61], see also [59,13]. For that purpose, the multiplier Lagrange vector λ is eliminated from the first equation of the system (75) using…”

“…The use of the combined approach alters the positive definitive nature of the structural stiffness matrix with the saddle equilibrium point and increases the number of unknowns of the microscopic BVP. The multiplier elimination procedure following [57] allows a unified multiple constraint treatment and has been successfully applied to enforce microscopic boundary conditions as demonstrated in [11,13,59]. In comparison with the direct constraint elimination method, the multiplier elimination method allows formulating the microscopic BVP in a path following strategy for problems involving instabilities [13].…”