Abstract
A new formulation of two-scale $$\hbox {FE}^2$$ analysis introduces symmetric stretch tensor as strain measure on macro level instead of asymmetric deformation gradient to determine boundary conditions on embedded microstructure. This significantly reduces computational cost of boundary conditions related sensitivity analysis of microstructure and with it the evaluation of local macroscopic stress tensors and tangent matrices. Various $$\hbox {FE}^2$$ formulations with isogeometric and standard finite element microanalysis are tested for consistency, accuracy and numerical efficiency on numerical homogenisation examples. Objective performance comparison of different $$\hbox {FE}^2$$ formulations is enabled with automation of all procedures in symbolic code generation system AceGen. The results obtained in numerical examples show reduced computational cost of the new $$\hbox {FE}^2$$ formulation without loss of accuracy and comparable numerical efficiency of higher order isogeometric and standard $$\hbox {FE}^2$$ formulations.
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