A multilevel finite element method (FE2) to describe the response of highly non-linear structures using generalized continua Original Research Article

A general method called FE2 has been introduced which consists in describing the behavior of heterogeneous structures using a multiscale finite element model. Instead of trying to build differential systems to establish a stress-strain relation at the macroscale, a finite element computation of the representative volume element is carried out simultaneously. Doing so does not require any constitutive equations to be written at the macroscopic scale: all non-linearities come directly from the microscale. In this paper, we describe how this method can be used in the context of generalized continua. For such continua, constitutive equations are very difficult to write, and a new set of material is difficult to fit to experimental data. The use of FE2 models bypasses this problem because no analytical equation is needed at the macroscale. An academic application is presented to show that generalized continua are necessary when the size of the heterogeneities increases, and that FE2 models behave well compared to a reference solution.