B. Ganapathysubramanian and N. Zabaras, A non-linear dimension reduction methodology for generating data-driven stochastic input models, Journal of Computational Physics, vol.227, issue.13, pp.6612-6637, 2008.
DOI : 10.1016/j.jcp.2008.03.023

O. Balima, Y. Favennec, and D. Petit, Model Reduction for Heat Conduction with Radiative Boundary Conditions using the Modal Identification Method, Numerical Heat Transfer, Part B: Fundamentals, vol.2, issue.2, pp.107-130, 2007.
DOI : 10.1109/TMAG.2002.804815

URL : https://hal.archives-ouvertes.fr/hal-00162009

D. Daescu and I. Navon, Efficiency of a POD-based reduced second-order adjoint model in 4D-Var data assimilation, International Journal for Numerical Methods in Fluids, vol.31, issue.6, pp.985-1004, 2007.
DOI : 10.1137/1.9780898719628

D. Ryckelynck, A priori hyperreduction method: an adaptive approach, Journal of Computational Physics, vol.202, issue.1, pp.346-366, 2005.
DOI : 10.1016/j.jcp.2004.07.015

D. Ryckelynck, Hyper-reduction of mechanical models involving internal variables, International Journal for Numerical Methods in Engineering, vol.1, issue.3, pp.75-89, 2009.
DOI : 10.1002/nme.2406

URL : https://hal.archives-ouvertes.fr/hal-00732247

F. Chinesta, R. Keunings, and A. Leygue, The Proper Generalized Decomposition for Advanced Numerical Simulations: a Primer. SpringerBriefs in applied sciences and technology, 2014.
DOI : 10.1007/978-3-319-02865-1

N. Cuaong, N. Veroy, K. Patera, and A. , Certified real-time solutin of parametrized partial differential equations Handbook of Material Modeling, pp.978-979, 2005.

A. Ammar, F. Chinesta, P. Diez, and A. Huerta, An error estimator for separated representations of highly multidimensional models, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.25-28, pp.25-28, 2010.
DOI : 10.1016/j.cma.2010.02.012

URL : https://hal.archives-ouvertes.fr/hal-01004991

P. Ladevèze and L. Chamoin, On the verification of model reduction methods based on the proper generalized decomposition, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.23-24, pp.23-242032, 2011.
DOI : 10.1016/j.cma.2011.02.019

P. Kerfriden, J. Ródenas, and S. Bordas, Certification of projection-based reduced order modelling in computational homogenisation by the constitutive relation error, International Journal for Numerical Methods in Engineering, vol.17, issue.4, pp.395-422, 2014.
DOI : 10.1007/s00466-013-0942-8

URL : https://hal.archives-ouvertes.fr/hal-00780840

J. De-almeida, A basis for bounding the errors of proper generalised decomposition solutions in solid mechanics, International Journal for Numerical Methods in Engineering, vol.137, issue.2, pp.961-984, 2013.
DOI : 10.1016/S0045-7825(96)01067-5

E. Arian, M. Fahl, and E. Sachs, Trust-region proper orthogonal decomposition for flow control, ICASE Report, 2000.

B. Michel and C. Laurent, Optimal control of the cylinder wake in the laminar regime by trust-region methods and pod reduced-order models, J Comput Phys, vol.227, pp.7813-7840, 2008.

D. Bui, M. Hamdaoui, D. Vuyst, and F. , POD-ISAT: An efficient POD-based surrogate approach with adaptive tabulation and fidelity regions for parametrized steady-state PDE discrete solutions, International Journal for Numerical Methods in Engineering, vol.86, issue.2, pp.648-671, 2013.
DOI : 10.1002/nme.3050

G. Rozza, D. Huynh, and A. Patera, Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations, Archives of Computational Methods in Engineering, vol.40, issue.11, pp.229-275, 2008.
DOI : 10.1016/j.crma.2003.09.023

L. Bris, C. Lelièvre, T. Maday, and Y. , Results and Questions on a Nonlinear Approximation Approach for Solving High-dimensional Partial Differential Equations, Constructive Approximation, vol.17, issue.2, pp.621-651, 2009.
DOI : 10.1137/1.9780898719574

URL : https://hal.archives-ouvertes.fr/inria-00336911

M. Barrault, Y. Maday, N. Nguyen, and A. Patera, An ???empirical interpolation??? method: application to efficient reduced-basis discretization of partial differential equations, Comptes Rendus Mathematique, vol.339, issue.9, pp.667-672, 2004.
DOI : 10.1016/j.crma.2004.08.006

URL : https://hal.archives-ouvertes.fr/hal-00021702

N. Nguyen, A. Patera, and J. Peraire, A ???best points??? interpolation method for efficient approximation of parametrized functions, International Journal for Numerical Methods in Engineering, vol.35, issue.4, pp.521-543, 2008.
DOI : 10.1016/j.crma.2004.08.006

M. Ortiz and L. Stainier, The variational formulation of viscoplastic constitutive updates, Computer Methods in Applied Mechanics and Engineering, vol.171, issue.3-4, pp.419-444, 1999.
DOI : 10.1016/S0045-7825(98)00219-9

URL : https://hal.archives-ouvertes.fr/hal-01007379

C. Miehe, Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation, International Journal for Numerical Methods in Engineering, vol.7, issue.11, pp.1285-1322, 2002.
DOI : 10.1051/m2an/197307R300331

N. Lahellec and P. Suquet, On the effective behavior of nonlinear inelastic composites: I. Incremental variational principles, Journal of the Mechanics and Physics of Solids, vol.55, issue.9, pp.1932-1963, 2007.
DOI : 10.1016/j.jmps.2007.02.003

URL : https://hal.archives-ouvertes.fr/hal-00214209

P. Ladevèze and D. Leguillon, Error Estimate Procedure in the Finite Element Method and Applications, SIAM Journal on Numerical Analysis, vol.20, issue.3, pp.485-509, 1983.
DOI : 10.1137/0720033

L. Gallimard, P. Ladevèze, and J. Pelle, ERROR ESTIMATION AND ADAPTIVITY IN ELASTOPLASTICITY, International Journal for Numerical Methods in Engineering, vol.94, issue.2, pp.189-217, 1996.
DOI : 10.1016/0045-7825(92)90057-Q

URL : https://hal.archives-ouvertes.fr/hal-01689591

P. Ladevèze and N. Moes, A Posteriori Constitutive Relation Error Estimators for Nonlinear Finite Element Analysis and Adaptive Control, Advances in Adaptive Computational Methods in Mechanics. Studies in Applied Mechanics, pp.231-256, 1998.
DOI : 10.1016/S0922-5382(98)80013-5

P. Ladevèze, B. Blaysat, and E. Florentin, Strict upper bounds of the error in calculated outputs of interest for plasticity problems, Computer Methods in Applied Mechanics and Engineering, vol.245, issue.246, pp.245-46194, 2012.
DOI : 10.1016/j.cma.2012.07.009

R. Bouclier, F. Louf, and L. Chamoin, Real-time validation of mechanical models coupling PGD and constitutive relation error, Computational Mechanics, vol.200, issue.23???24, pp.861-883, 2013.
DOI : 10.1016/j.cma.2011.02.019

URL : https://hal.archives-ouvertes.fr/hal-01580007

J. Pelle and D. Ryckelynck, An efficient adaptive strategy to master the global quality of viscoplastic analysis, Computers & Structures, vol.78, issue.1-3, pp.169-183, 2000.
DOI : 10.1016/S0045-7949(00)00107-3

M. Paraschivoiu, J. Peraire, and A. Patera, A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations, Symposium on Advances in Computational Mechanics, pp.289-31200086, 1997.
DOI : 10.1016/S0045-7825(97)00086-8

D. Wirtz and B. Haasdonk, Efficient a-posteriori error estimation for nonlinear kernel-based reduced systems, Systems & Control Letters, vol.61, issue.1, pp.203-211, 2012.
DOI : 10.1016/j.sysconle.2011.10.012

K. Karhunen, Über lineare methoden in der wahrscheinlichkeitsrechnung, Ann Academiae Scientiarum Fennicae, vol.37, pp.3-79, 1947.

L. Sirovich, Turbulence and the dynamics of coherent structures. I. Coherent structures, Quarterly of Applied Mathematics, vol.45, issue.3, pp.561-571, 1987.
DOI : 10.1090/qam/910462

R. Radovitzky and M. Ortiz, Error estimation and adaptive meshing in strongly nonlinear dynamic problems, Computer Methods in Applied Mechanics and Engineering, vol.172, issue.1-4, pp.203-240, 1999.
DOI : 10.1016/S0045-7825(98)00230-8

URL : https://thesis.library.caltech.edu/4386/1/Radovitzky_ra_1998.pdf

M. Biot, Mechanics of Incremental Deformation, Journal of Applied Mechanics, vol.32, issue.4, 1965.
DOI : 10.1115/1.3627365

H. Ziegler, Some Extremum Principles in Irreversible Thermodynamics with Applications to Continuum Mechanics, Progress in Solid Mechanics, 1963.

P. Germain, Q. Nguyen, and P. Suquet, Continuum Thermodynamics, Journal of Applied Mechanics, vol.50, issue.4b, pp.1010-1020, 1983.
DOI : 10.1115/1.3167184

URL : https://hal.archives-ouvertes.fr/hal-00105630

B. Halphen and Q. Nguyen, Generalized standard materials, J De Mecanique, vol.14, issue.1, pp.39-63, 1975.

J. Lemaitre and J. Chaboche, Mecanique des Materiaux Solides, 1985.

D. Ryckelynck, Hyper-reduction of mechanical models involving internal variables, International Journal for Numerical Methods in Engineering, vol.1, issue.3, pp.75-89, 2009.
DOI : 10.1002/nme.2406

URL : https://hal.archives-ouvertes.fr/hal-00732247

J. Besson, G. Cailletaud, J. Chaboche, and S. Forest, Non-linear mechanics of materials, Solid Mechanics and Its Applications, p.2010, 2009.
DOI : 10.1007/978-90-481-3356-7

F. Fritzen and T. Bhlke, Nonlinear homogenization using the nonuniform transformation field analysis, PAMM, vol.11, issue.1, pp.519-522, 2011.
DOI : 10.1002/pamm.201110250

F. Fritzen and M. Leuschner, Reduced basis hybrid computational homogenization based on a mixed incremental formulation, Computer Methods in Applied Mechanics and Engineering, vol.260, issue.0, pp.143-154, 2013.
DOI : 10.1016/j.cma.2013.03.007