Variational formulation and monolithic solution of computational homogenization methods

In this contribution, we derive a consistent variational formulation for computational homogenization methods and show that traditional FE${\hspace{0.17em}}^2$ and IGA${\hspace{0.17em}}^2$ approaches are special discretization and solution techniques of this most general framework. This allows us to enhance dramatically the numerical analysis as well as the solution of the arising algebraic system. In particular, we expand the dimension of the continuous system, discretize the higher dimensional problem consistently and apply afterwards a discrete null-space matrix to remove the additional dimensions. A benchmark problem, for which we can obtain an analytical solution, demonstrates the superiority of the chosen approach aiming to reduce the immense computational costs of traditional FE${\hspace{0.17em}}^2$ and IGA${\hspace{0.17em}}^2$ formulations to a fraction of the original requirements. Finally, we demonstrate a further reduction of the computational costs for the solution of general nonlinear problems.