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In this contribution, we derive a consistent variational formulation forcomputational homogenization methods and show that traditional FE2 and IGA2approaches are special discretization and solution techniques of this mostgeneral framework. This allows us to enhance dramatically the numericalanalysis as well as the solution of the arising algebraic system. Inparticular, we expand the dimension of the continuous system, discretize thehigher dimensional problem consistently and apply afterwards a discretenull-space matrix to remove the additional dimensions. A benchmark problem, forwhich we can obtain an analytical solution, demonstrates the superiority of thechosen approach aiming to reduce the immense computational costs of traditionalFE2 and IGA2 formulations to a fraction of the original requirements. Finally,we demonstrate a further reduction of the computational costs for the solutionof general non-linear problems.

Variational formulation and monolithic solution of computational homogenization methods