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In this paper we address three aspects of nonlinear computationalhomogenization of elastic solids by two-scale finite element methods. First, wepresent a nonlinear formulation of the finite element heterogeneous multiscalemethod FE-HMM in a Lagrangean formulation that covers geometrical nonlinearityand, more general, hyperelasticity. Second, a-priori estimates of FE-HMM, whichexist so far only for the fully linear elastic case in solid mechanics, areassessed in the regime of nonlinear elasticity. The measured convergence ratesagree fairly well with those of the fully linear regime. Third, we revise thestandard solution algorithm of FE$^2$ which is a staggered scheme in terms of anested loop embedding the full solution of the micro problem into one macrosolution iteration step. We demonstrate that suchlike staggered scheme, whichis typically realized by a nested two-level Newton algorithm, can safely andefficiently be replaced by direct alternations between micro and macroiterations. The novel algorithmic structure is exemplarily detailed for theproposed nonlinear FE-HMM, its efficiency is substantiated by a considerablespeedup in numerical tests.

A Nonlinear Finite Element Heterogeneous Multiscale Method for the  Homogenization of Hyperelastic Solids and a Novel Staggered Two-Scale  Solution Algorithm