High Performance Reduced Order Modeling Techniques Based on Optimal Energy Quadrature: Application to Geometrically Non-linear Multiscale Inelastic Material Modeling

  • Manuel Caicedo
  • Javier L. Mroginski
  • Sebastian Toro
  • Marcelo Raschi
  • Alfredo Huespe
  • Javier OliverEmail author
Original Paper


A High-Performance Reduced-Order Model (HPROM) technique, previously presented by the authors in the context of hierarchical multiscale models for non linear-materials undergoing infinitesimal strains, is generalized to deal with large deformation elasto-plastic problems. The proposed HPROM technique uses a Proper Orthogonal Decomposition procedure to build a reduced basis of the primary kinematical variable of the micro-scale problem, defined in terms of the micro-deformation gradient fluctuations. Then a Galerkin-projection, onto this reduced basis, is utilized to reduce the dimensionality of the micro-force balance equation, the stress homogenization equation and the effective macro-constitutive tangent tensor equation. Finally, a reduced goal-oriented quadrature rule is introduced to compute the non-affine terms of these equations. Main importance in this paper is given to the numerical assessment of the developed HPROM technique. The numerical experiments are performed on a micro-cell simulating a randomly distributed set of elastic inclusions embedded into an elasto-plastic matrix. This micro-structure is representative of a typical ductile metallic alloy. The HPROM technique applied to this type of problem displays high computational speed-ups, increasing with the complexity of the finite element model. From these results, we conclude that the proposed HPROM technique is an effective computational tool for modeling, with very large speed-ups and acceptable accuracy levels with respect to the high-fidelity case, the multiscale behavior of heterogeneous materials subjected to large deformations involving two well-separated scales of length.

List of Symbols


Number of orthonormal reduced basis for the micro-gradient deformation fluctuation space

\(n_{\varphi }\)

Number of orthonormal reduced basis for the micro-elastic free energy space


Number of quadrature points of the HFFEM (Gauss point number)


Number of quadrature points defining the ROQ rule


Total number of snapshots taken from the micro-cell sampling program

\({[}\varvec{\chi }{]}_{\tilde{\varvec{F}}_{\mu }}\)

Matrix of snapshots of deformation gradient fluctuations

\({[}\varvec{\chi }{]}_{\varphi _{\mu }}\)

Matrix of elastic energy snapshots

\(\{{\varvec{\Psi }}\}\)

Reduced order basis of the deformation gradient fluctuations

\(\{\varvec{\Phi }\}\)

Reduced order basis of the elastic energy



Proper Orthogonal Decomposition


Singular Value Decomposition


High-Performance Reduced Order Model


Hyper-Reduced Order Model.


High-Fidelity Finite Element Model (model based on the original high-order finite element mesh)


Reduced Order Model


Reduced Optimal Quadrature


Initial Boundary Value Problem


Representative Volume Element



The authors acknowledge the financial support from the European Research Council through the following grants: 1) European Union Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement N. 320815 (ERC Advanced Grant Project Advanced tools for computational design of engineering materials COMP-DES-MAT) and 2) Proof of Concept: ERC-2017-PoC 779611 (Computational catalog of multiscale materials: a plugin library for industrial finite elements codes, CATALOG). Also, second and fourth authors acknowledge the financial support from CONICET and ANPCyT (grants PIP 2013-2015 631 and PICT 2014-3372). The authors would like also to acknowledge the support of Dr. Joaquin Hernández, from CIMNE, on the algorithmic and technical aspects of the reduced order integration methods used in this work.

Compliance with Ethical Standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


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Copyright information

© CIMNE, Barcelona, Spain 2018

Authors and Affiliations

  1. 1.CIMNE – Centre Internacional de Metodes Numerics en Enginyeria, Campus Nord UPCBarcelonaSpain
  2. 2.CONICET, Argentine Council for Science and TechnologyResistenciaArgentina
  3. 3.CIMEC-UNL-CONICETSanta FeArgentina
  4. 4.E.T.S d’Enginyers de Camins, Canals i PortsTechnical University of Catalonia (BarcelonaTech), Campus Nord UPCBarcelonaSpain

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