Comparison of reduced order homogenization techniques: pRBMOR, NUTFA and MxTFA

The introduction of composites in engineering applications led to a need for tools that can predict the mechanical response with account for the heterogeneities in the materials in order to safely design complex structures. These predictions are required to be sufficiently accurate yet computationally inexpensive, especially when dealing with nonlinear materials: for these the amount of internal variables and the computing times can both become prohibitively high. Here, three model order reduction homogenization methods are compared in detail: the potential-based Reduced Basis Model Order Reduction (pRBMOR), the NonUniform Transformation Field Analysis and the Mixed Transformation Field Analysis (MxTFA). All methods are developed in the framework of the Nonuniform TFA, initially introduced by Michel and Suquet. Two of the illustrated methods, the pRBMOR and the MxTFA, deviate from the TFA of Michel and Suquet in that they are based on mixed variational formulations. The comparison will investigate differences and similarities of the techniques in terms of the reduced degrees of freedom and their evolution law, the storage requirements and the accuracy of the effective constitutive behavior. Numerical tests on three-dimensional periodic composites under complex normal and shear loading paths will show the performances of the techniques.