, Volume 53, Issue 6, pp 1291–1312 | Cite as

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

  • F. Covezzi
  • S. de Miranda
  • F. Fritzen
  • S. Marfia
  • E. Sacco
Novel Computational Approaches to Old and New Problems in Mechanics


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.


Homogenization Viscoplastic composites Mixed formulation Transformation Field Analysis 



Financial supports by the Italian Ministry of Education, University and Research—MIUR are gratefully acknowledged: PRIN2015 Advanced mechanical modeling of new materials and structures for the solution of 2020 Horizon challenges   prot. 2015JW9NJT_018, PRIN2015 Multi-scale mechanical models for the design and optimization of micro-structured smart materials and metamaterials   prot. 2015LYYXA8_002. Parts of this work were effected by the Emmy-Noether-Group EMMA supported through Grant DFG-FR2702/6 within the Emmy-Noether program of the Deutsche Forschungsgemeinschaft (DFG) and within the scope of the scientific network CoSiMOR (DFG Grant FR2702/7). The authors gratefully acknowledge the financial and personal support of the DFG.


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • F. Covezzi
    • 1
  • S. de Miranda
    • 1
  • F. Fritzen
    • 2
  • S. Marfia
    • 3
  • E. Sacco
    • 4
  1. 1.DICAMUniversity of BolognaBolognaItaly
  2. 2.Institute of Applied Mechanics (CE)University of StuttgartStuttgartGermany
  3. 3.DICeMUniversity of Cassino and of Southern LazioCassinoItaly
  4. 4.DiStUniversity of Naples Federico IINaplesItaly

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