, 121 6.2 CRE functional for nonlinear material behaviors, p.122

. .. Cre-for-damage-problems, 125 6.3.1 Example : damage problem on a plate with circular hole, p.127

, Parametrized reference model with damage behavior

, Example : nonlinear problem on a plate with circular hole using PGD, Procedure for reference model with damage behaviour . . . 130 6.5.1 Step 1: non-damage stage

, We think the coupling between IGA, PGD, and CRE tools is a valuable approach, making scientific advances in the field, and which paves the way for further studies. Here, we thus propose to couple PGD with IGA for nonlinear damage problems with parametrized Bibliography

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, pp.1872-1880, 2010.
URL : https://hal.archives-ouvertes.fr/hal-01004991

F. Auricchio, F. Calabrò, T. J. Hughes, A. Reali, and G. Sangalli, A simple algorithm for obtaining nearly optimal quadrature rules for NURBS-based isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, pp.15-27, 2012.

P. Allier, L. Chamoin, and P. Ladevèze, Towards simplified and optimized a posteriori error estimation using PGD reduced models, International Journal for Numerical Methods in Engineering, vol.113, issue.6, pp.967-998, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01584380

O. Allix and J. Deü, Delayed-Damage Modelling for Fracture Prediction of Laminated Composites under Dynamic Loading, Engineering Transactions, vol.45, issue.1, pp.29-46, 1997.

V. and S. S. Gautam, Varying-order NURBS discretization: An accurate and efficient method for isogeometric analysis of large deformation contact problems, 2019.

I. Alfaro, D. González, S. Zlotnik, P. Díez, E. Cueto et al., An error estimator for real-time simulators based on model order reduction. Advanced Modeling and Simulation in Engineering Sciences, vol.2, p.30, 2015.

A. Ammar, A. Huerta, F. Chinesta, E. Cueto, and A. Leygue, Parametric solutions involving geometry: A step towards efficient shape optimization, Computer Methods in Applied Mechanics and Engineering, vol.268, pp.178-193, 2014.

J. P. Almeida, A basis for bounding the errors of proper generalised decomposition solutions in solid mechanics, International Journal for Numerical Methods in Engineering, vol.94, issue.10, pp.961-984, 2013.

J. P. Almeida and E. A. Maunder, Equilibrium Finite Element Formulations, 2017.

A. Ammar, B. Mokdad, F. Chinesta, R. Keunings-;-a.-ammar, M. Normandin et al., A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids, Non-Incremental Strategies Based on Separated Representations : Applications in Computational Rheology, vol.139, pp.671-695, 2006.
URL : https://hal.archives-ouvertes.fr/hal-01004909

M. Ainsworth and J. T. Oden, A unified approach to a posteriori error estimation using element residual methods, Numerische Mathematik, vol.65, issue.1, pp.23-50, 1993.

M. Ainsworth and J. T. Oden, A posteriori error estimation in finite element analysis, Computer Methods in Applied Mechanics and Engineering, vol.142, issue.1, pp.1-88, 1997.

Y. Bazilevs, L. Beirão-da, J. A. Veiga, T. J. Cottrell, G. Hughes et al., Isogeometric analysis: Approximation, stability and error estimates for h-refined meshes, Mathematical Models and Methods in Applied Sciences, vol.16, issue.07, pp.1031-1090, 2006.

D. J. Benson, Y. Bazilevs, M. C. Hsu, and T. J. Hughes, Isogeometric shell analysis: The Reissner-Mindlin shell, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.5, pp.276-289, 2010.

D. J. Benson, Y. Bazilevs, M. C. Hsu, and T. J. Hughes, A large deformation, rotation-free, isogeometric shell, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.13, pp.1367-1378, 2011.

Y. Bazilevs, V. M. Calo, T. J. Hughes, and Y. Zhang, Isogeometric fluid-structure interaction: theory, algorithms, and computations. Computational Mechanics, vol.43, pp.3-37, 2008.

L. Beirão-da-veiga, A. Buffa, C. Lovadina, M. Martinelli, and G. Sangalli, An isogeometric method for the Reissner-Mindlin plate bending problem, Computer Methods in Applied Mechanics and Engineering, pp.45-53, 2012.

L. Beirão-da-veiga, A. Buffa, J. Rivas, and G. Sangalli, Some estimates for h-p-k-refinement in Isogeometric Analysis, Numerische Mathematik, vol.118, issue.2, pp.271-305, 2011.

R. Bouclier, T. Elguedj, and A. Combescure, An isogeometric locking-free NURBS-based solid-shell element for geometrically nonlinear analysis, International Journal for Numerical Methods in Engineering, vol.101, issue.10, pp.774-808, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01978232

D. J. Benson, S. Hartmann, Y. Bazilevs, M. C. Hsu, and T. J. Hughes, Blended isogeometric shells, Computer Methods in Applied Mechanics and Engineering, vol.255, pp.133-146, 2013.

S. Boyaval, Reduced-Basis Approach for Homogenization beyond the Periodic Setting, Multiscale Modeling & Simulation, vol.7, issue.1, pp.466-494, 2008.
URL : https://hal.archives-ouvertes.fr/inria-00132763

D. Braess, V. Pillwein, and J. Schöberl, Equilibrated residual error estimates are p-robust, Computer Methods in Applied Mechanics and Engineering, vol.198, issue.13, pp.1189-1197, 2009.

R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numerica, vol.10, pp.1-102, 2001.

R. Bank and R. Smith, A Posteriori Error Estimates Based on Hierarchical Bases, SIAM Journal on Numerical Analysis, vol.30, issue.4, pp.921-935, 1993.

I. Babuska and T. Strouboulis, The Finite Element Method and its Reliability. Numerical Mathematics and Scientific Computation, 2001.

M. J. Borden, M. A. Scott, J. A. Evans, and T. J. Hughes, Isogeometric finite element data structures based on Bézier extraction of NURBS, International Journal for Numerical Methods in Engineering, vol.87, issue.1-5, pp.15-47, 2011.

I. Babu?ka, T. Strouboulis, C. S. Upadhyay, S. K. Gangaraj, and K. Copps, Validation of a posteriori error estimators by numerical approach, International Journal for Numerical Methods in Engineering, vol.37, issue.7, pp.1073-1123, 1994.

A. Buffa, G. Sangalli, and R. Vázquez, Isogeometric analysis in electromagnetics: B-splines approximation, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.17, pp.1143-1152, 2010.

F. Chinesta, A. Ammar, and E. Cueto, Recent Advances and New Challenges in the Use of the Proper Generalized Decomposition for Solving Multidimensional Models, Archives of Computational Methods in Engineering, vol.17, issue.4, pp.327-350, 2010.
URL : https://hal.archives-ouvertes.fr/hal-01007235

T. Canuto and C. Kozubek, A fictitious domain approach to the numerical solution of PDEs in stochastic domains, Numerische Mathematik, vol.107, issue.2, p.257, 2007.

R. Cottereau, P. Díez, and A. Huerta, Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free method, Computational Mechanics, vol.44, issue.4, pp.533-547, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00707098

A. Chemin, T. Elguedj, and A. Gravouil, Isogeometric local h-refinement strategy based on multigrids, Finite Elements in Analysis and Design, vol.100, pp.77-90, 2015.

C. Carstensen and S. Funken, Fully Reliable Localized Error Control in the FEM, SIAM Journal on Scientific Computing, vol.21, issue.4, pp.1465-1484, 1999.

J. Cottrell, T. J. Hughes, and Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA, 2009.

F. Chinesta, A. Huerta, G. Rozza, and K. Willcox, Model Reduction Methods, Encyclopedia of Computational Mechanics Second Edition, pp.1-36, 2017.
URL : https://hal.archives-ouvertes.fr/hal-00110008

F. Chinesta, R. Keunings, and A. Leygue, The Proper Generalized Decomposition for Advanced Numerical Simulations: A Primer. SpringerBriefs in Applied Sciences and Technology, 2014.

L. Chamoin and P. Ladevèze, Bounds on history-dependent or independent local quantities in viscoelasticity problems solved by approximate methods, International Journal for Numerical Methods in Engineering, vol.71, issue.12, pp.1387-1411, 2007.
URL : https://hal.archives-ouvertes.fr/hal-01580945

L. Chamoin and P. Ladevèze, A non-intrusive method for the calculation of strict and efficient bounds of calculated outputs of interest in linear viscoelasticity problems, Computer Methods in Applied Mechanics and Engineering, vol.197, issue.9, pp.994-1014, 2008.
URL : https://hal.archives-ouvertes.fr/hal-01580943

L. Chamoin and P. Ladevèze, Robust control of PGD-based numerical simulations, European Journal of Computational Mechanics, vol.21, issue.3-6, pp.195-207, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01581354

, Separated Representations and PGD-Based Model Reduction: Fundamentals and Applications. CISM International Centre for Mechanical Sciences, 2014.

F. Chinesta, A. Leygue, F. Bordeu, J. V. Aguado, E. Cueto et al., PGD-Based Computational Vademecum for Efficient Design, Optimization and Control. Archives of Computational Methods in Engineering, vol.20, issue.1, pp.31-59, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01515083

F. Chinesta, P. Ladevèze, and E. Cueto, A Short Review in Model Order Reduction Based on Proper Generalized Decomposition, Archives of Computational Methods in Engineering, vol.18, issue.4, pp.395-404, 2011.
URL : https://hal.archives-ouvertes.fr/hal-01004940

A. Courard, D. Néron, P. Ladevèze, and L. Ballere, Integration of PGDvirtual charts into an engineering design process, Computational Mechanics, vol.57, issue.4, pp.637-651, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01647844

M. Chevreuil, A. Nouy, and E. Safatly, A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties, Computer Methods in Applied Mechanics and Engineering, vol.255, pp.255-274, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00733739

L. Chamoin, F. Pled, P. Allier, and P. Ladevèze, A posteriori error estimation and adaptive strategy for PGD model reduction applied to parametrized linear parabolic problems, Computer Methods in Applied Mechanics and Engineering, vol.327, pp.118-146, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01584532

J. A. Cottrell, A. Reali, Y. Bazilevs, and T. J. Hughes, Isogeometric analysis of structural vibrations, Computer Methods in Applied Mechanics and Engineering, vol.195, issue.41, pp.5257-5296, 2006.
URL : https://hal.archives-ouvertes.fr/hal-01516398

C. De-boor, ;. L. Dalcin, N. Collier, P. Vignal, A. M. Côrtes et al., PetIGA: A framework for high-performance isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, vol.308, pp.151-181, 2001.

M. R. Dörfel, B. Jüttler, and B. Simeon, Adaptive isogeometric analysis by local h-refinement with T-splines, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.5, pp.264-275, 2010.

X. Deng, A. Korobenko, J. Yan, and Y. Bazilevs, Isogeometric analysis of continuum damage in rotation-free composite shells, Computer Methods in Applied Mechanics and Engineering, vol.284, pp.349-372, 2015.

L. De-lorenzis, P. Wriggers, and G. Zavarise, A mortar formulation for 3d large deformation contact using NURBS-based isogeometric analysis and the augmented Lagrangian method, Computational Mechanics, vol.49, issue.1, pp.1-20, 2012.

P. Destuynder and B. Métivet, Explicit Error Bounds in a Conforming Finite Element Method, Mathematics of Computation, vol.68, issue.228, pp.1379-1396, 1999.

J. A. Evans, Y. Bazilevs, I. Babu?ka, and T. J. Hughes, n-Widths, sup-infs, and optimality ratios for the k-version of the isogeometric finite element method, Computer Methods in Applied Mechanics and Engineering, vol.198, issue.21, pp.1726-1741, 2009.

T. Elguedj, Y. Bazilevs, V. M. Calo, and T. J. Hughes, B-bar and Fbar projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements, Computer Methods in Applied Mechanics and Engineering, vol.197, pp.2732-2762, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00457010

R. Echter, B. Oesterle, and M. Bischoff, A hierarchic family of isogeometric shell finite elements, Computer Methods in Applied Mechanics and Engineering, vol.254, pp.170-180, 2013.

A. Ern and M. Vohralík, A Posteriori Error Estimation Based on Potential and Flux Reconstruction for the Heat Equation, SIAM Journal on Numerical Analysis, vol.48, issue.1, pp.198-223, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00383692

D. Fußeder, B. Simeon, and A. V. Vuong, Fundamental aspects of shape optimization in the context of isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, vol.286, pp.313-331, 2015.

L. Gallimard, A constitutive relation error estimator based on tractionfree recovery of the equilibrated stress, International Journal for Numerical Methods in Engineering, vol.78, issue.4, pp.460-482, 2009.
URL : https://hal.archives-ouvertes.fr/hal-01689801

H. Gómez, V. M. Calo, Y. Bazilevs, and T. J. Hughes, Isogeometric analysis of the Cahn-Hilliard phase-field model, Computer Methods in Applied Mechanics and Engineering, vol.197, issue.49, pp.4333-4352, 2008.

D. Großmann, B. Jüttler, H. Schlusnus, J. Barner, and A. Vuong, Isogeometric simulation of turbine blades for aircraft engines, Computer Aided Geometric Design, vol.29, issue.7, pp.519-531, 2012.

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

M. B. Giles and E. Süli, Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, Acta Numerica, p.11, 2002.

J. Gu, T. Yu, L. Van-lich, T. Nguyen, and T. Bui, Adaptive multipatch isogeometric analysis based on locally refined B-splines, Computer Methods in Applied Mechanics and Engineering, vol.339, pp.704-738, 2018.

T. J. Hughes, J. A. Cottrell, and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement
URL : https://hal.archives-ouvertes.fr/hal-01513346

, Computer Methods in Applied Mechanics and Engineering, vol.194, issue.39, pp.4135-4195, 2005.

J. Haslinger and R. Mäkinen, Introduction to shape optimization. Theory, approximation, and computation, vol.01, 2003.

B. Halphen and Q. Nguyen, Sur les Matériaux Standard Généralisés, Journal de Mécanique, vol.14, pp.39-63, 1975.

T. J. Hughes, A. Reali, and G. Sangalli, Efficient quadrature for NURBS-based isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.5, pp.301-313, 2010.

S. Hosseini, J. J. Remmers, C. V. Verhoosel, and R. Borst, Propagation of delamination in composite materials with isogeometric continuum shell elements, International Journal for Numerical Methods in Engineering, vol.102, issue.3-4, pp.159-179, 2015.

K. A. Johannessen and M. Kumar, file = Full Text PDF:/Users/phuong/Zotero/storage/5V33BDDK/Johannessen et al. -2015 -Divergence-conforming discretization for Stokes pr.pdf:application/pdf;Snapshot:/Users/phuong/Zotero/storage/B9B4QWRL/2498413.htm Kvamsdal, T. Divergence-conforming discretization for Stokes problem on locally refined meshes using LR B-splines, pp.38-70, 2015.

J. Kiendl, K. U. Bletzinger, J. Linhard, and R. Wüchner, Isogeometric shell analysis with Kirchhoff-Love elements, Computer Methods in Applied Mechanics and Engineering, vol.198, issue.49, pp.3902-3914, 2009.

P. Kagan, A. Fischer, and P. , Bar-Yoseph. New B-Spline Finite Element approach for geometrical design and mechanical analysis, International Journal for Numerical Methods in Engineering, vol.41, issue.3, pp.435-458, 1998.

M. Kumar, T. Kvamsdal, and K. A. Johannessen, Superconvergent patch recovery and a posteriori error estimation technique in adaptive isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, vol.316, pp.1086-1156, 2017.

R. Kruse, N. Nguyen-thanh, L. D. Lorenzis, and T. J. Hughes, Isogeometric collocation for large deformation elasticity and frictional contact problems, Computer Methods in Applied Mechanics and Engineering, vol.296, pp.73-112, 2015.

S. K. Kleiss and S. K. Tomar, Guaranteed and sharp a posteriori error estimates in isogeometric analysis, Computers & Mathematics with Applications, vol.70, issue.3, pp.167-190, 2015.

G. Kuru, C. V. Verhoosel, K. G. Van-der-zee, and E. H. Van-brummelen, Goal-adaptive Isogeometric Analysis with hierarchical splines

, Computer Methods in Applied Mechanics and Engineering, vol.270, pp.270-292, 2014.

P. Ladevèze, Nonlinear Computational Structural Mechanics: New Approaches and Non-Incremental Methods of Calculation. Mechanical Engineering Series, 1999.

P. Ladevèze, Constitutive relation errors for F.E. analysis considering (visco-) plasticity and damage, International Journal for Numerical Methods in Engineering, vol.52, issue.5-6, pp.527-542, 2001.

P. Ladevèze, Strict upper error bounds on computed outputs of interest in computational structural mechanics, Computational Mechanics, vol.42, issue.2, pp.271-286, 2008.

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, pp.2032-2047, 2011.

P. Ladevèze and L. Chamoin, Toward guaranteed PGD-reduced models, Bytes and Science, 2012.

P. Ladevèze and L. Chamoin, The constitutive relation error method: A general verification tool. verifying calculations -forty years on an overview of classical verification techniques for fem simulations, pp.59-94, 2015.

P. Ladevèze, L. Chamoin, and . Florentin, A new non-intrusive technique for the construction of admissible stress fields in model verification, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.9, pp.766-777, 2010.

Y. Liang, X. Cheng, Z. Li, and J. Xiang, Multi-objective robust airfoil optimization based on non-uniform rational B-spline (NURBS) representation, Science China Technological Sciences, vol.53, issue.10, pp.2708-2717, 2010.

S. Lipton, J. A. Evans, Y. Bazilevs, T. Elguedj, and T. J. Hughes, Robustness of isogeometric structural discretizations under severe mesh distortion, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.5, pp.357-373, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00457008

J. Lepine, F. Guibault, J. Trépanier, and F. Pepin, Optimized nonuniform rational b-spline geometrical representation for aerodynamic design of wings, AIAA Journal, vol.39, pp.2033-2041, 2001.

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.

P. Ladevèze and E. A. Maunder, A general method for recovering equilibrating element tractions, Computer Methods in Applied Mechanics and Engineering, vol.137, issue.2, pp.111-151, 1996.

P. Ladevèze and N. Moës, A new a posteriori error estimation for nonlinear time-dependent finite element analysis, Computer Methods in Applied Mechanics and Engineering, vol.157, issue.1, pp.45-68, 1998.

P. Ladevèze, N. Moës, and B. Douchin, Constitutive relation error estimators for (visco)plastic finite element analysis with softening, Computer Methods in Applied Mechanics and Engineering, vol.176, issue.1, pp.247-264, 1999.

P. Ladevèze and J. P. Pelle, Mastering Calculations in Linear and Nonlinear Mechanics, Mechanical Engineering Series, 2005.

P. Ladevèze, D. Passieux, J. , and C. Néron, The LATIN multiscale computational method and the Proper Generalized Decomposition, Computer Methods in Applied Mechanics and Engineering, vol.199, pp.1287-1296, 2010.

K. Li and X. Qian, Isogeometric analysis and shape optimization via boundary integral, Computer-Aided Design, vol.43, issue.11, pp.1427-1437, 2011.

P. Ladevèze, . Ph, and . Rougeot, New advances on a posteriori error on constitutive relation in f.e. analysis, Computer Methods in Applied Mechanics and Engineering, vol.150, issue.1, pp.239-249, 1997.

T. Lassila and G. Rozza, Parametric free-form shape design with PDE models and reduced basis method, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.23, pp.1583-1592, 2010.

E. Laporte and P. L. Tallec, Numerical Methods in Sensitivity Analysis and Shape Optimization. Modeling and Simulation in Science, Engineering and Technology, 2003.

A. Leygue and E. Verron, A First Step Towards the Use of Proper General Decomposition Method for Structural Optimization, Archives of Computational Methods in Engineering, vol.17, issue.4, pp.465-472, 2010.
URL : https://hal.archives-ouvertes.fr/hal-01004831

L. Lorenzis, P. Wriggers, T. J. Hughes, ;. S. Li, Z. Wu et al., An isogeometric boundary element reanalysis framework based on proper generalized decomposition, 8th International Conference on Mechanical and Aerospace Engineering (ICMAE), vol.37, pp.272-280, 2014.

L. Machiels, Y. Maday, and A. T. Patera, A "flux-free" nodal Neumann subproblem approach to output bounds for partial differential equations

, Comptes Rendus de l'Académie des Sciences -Series I -Mathematics, vol.330, issue.3, pp.249-254, 2000.

A. Manzoni, A. Quarteroni, and G. Rozza, Shape optimization for viscous flows by reduced basis methods and free-form deformation, International Journal for Numerical Methods in Fluids, vol.70, issue.5, pp.646-670, 2012.

A. Manzoni, F. Salmoiraghi, and L. Heltai, Reduced Basis Isogeometric Methods (RB-IGA) for the real-time simulation of potential flows about parametrized NACA airfoils, 2015.

V. P. Nguyen, C. Anitescu, S. P. Bordas, and T. Rabczuk, Isogeometric analysis: An overview and computer implementation aspects, Mathematics and Computers in Simulation, vol.117, pp.89-116, 2015.

A. Nouy, M. Chevreuil, and E. Safatly, Fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains, Computer Methods in Applied Mechanics and Engineering, pp.3066-3082, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00662564

N. C. Nguyen, A multiscale reduced-basis method for parametrized elliptic partial differential equations with multiple scales, Journal of Computational Physics, vol.227, issue.23, pp.9807-9822, 2008.

A. Nouy, A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.23, pp.1603-1626, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00455635

A. Nouy and F. Pled, A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities, ESAIM: Mathematical Modelling and Numerical Analysis, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01507489

H. Nguyen-xuan, T. Hoang, and V. P. Nguyen, An isogeometric analysis for elliptic homogenization problems, Computers & Mathematics with Applications, vol.67, issue.9, pp.1722-1741, 2014.

J. T. Oden, J. Fish, C. Johnson, A. Laub, T. Srolovitz et al., Simulation-based engineering science, vol.01, 2006.

E. Pruliere, F. Chinesta, and A. Ammar, On the deterministic solution of multidimensional parametric models using the Proper Generalized Decomposition, Mathematics and Computers in Simulation, vol.81, issue.4, pp.791-810, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00704427

F. Pled, L. Chamoin, and P. Ladevèze, On the techniques for constructing admissible stress fields in model verification: Performances on engineering examples, International Journal for Numerical Methods in Engineering, vol.88, issue.5, pp.409-441, 2011.
URL : https://hal.archives-ouvertes.fr/hal-01056705

N. Parés, P. Díez, and A. Huerta, Subdomain-based flux-free a posteriori error estimators, Computer Methods in Applied Mechanics and Engineering, vol.195, issue.4, pp.297-323, 2006.

S. Prudhomme and J. T. Oden, On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors, Computer Methods in Applied Mechanics and Engineering, vol.176, issue.1, pp.313-331, 1999.

M. Paraschivoiu, A. T. Peraire, and J. Patera, A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations, Computer Methods in Applied Mechanics and Engineering, vol.150, issue.1, pp.289-312, 1997.

W. Prager and J. L. Synge, Approximation in elasticity based on the concept of function space, Quarterly of Applied Mathematics, vol.5, issue.3, pp.241-269, 1947.

L. Piegl and W. Tiller, The NURBS Book. Monographs in Visual Communication, 1995.

V. Rey, P. Gosselet, and C. Rey, Study of the strong prolongation equation for the construction of statically admissible stress fields: Implementation and optimization, Computer Methods in Applied Mechanics and Engineering, vol.268, pp.82-104, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00862622

G. Rozza, Reduced Basis Approximation and Error Bounds for Potential Flows in Parametrized Geometries, Communications in Computational Physics, vol.9, issue.1, pp.1-48, 2011.

R. N. Simpson, S. P. Bordas, J. Trevelyan, and T. Rabczuk, A two-dimensional Isogeometric Boundary Element Method for elastostatic analysis, Computer Methods in Applied Mechanics and Engineering, pp.87-100, 2012.

D. Schillinger, L. Dedè, M. A. Scott, J. A. Evans, M. J. Borden et al., An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces, Computer Methods in Applied Mechanics and Engineering, pp.116-150, 2012.

R. Sevilla, S. Fernández-méndez, and A. Huerta, 3d NURBS-enhanced finite element method (NEFEM), International Journal for Numerical Methods in Engineering, vol.88, issue.2, pp.103-125, 2011.

A. Sobester and A. J. Keane, Airfoil design via cubic splines -Ferguson's curves revisited, pp.1-15, 2007.

Y. Seo, H. Kim, and S. Youn, Shape optimization and its extension to topological design based on isogeometric analysis, International Journal of Solids and Structures, vol.47, issue.11, pp.1618-1640, 2010.

M. A. Scott, R. N. Simpson, J. A. Evans, S. Lipton, S. P. Bordas et al., Isogeometric boundary element analysis using unstructured T-splines, Computer Methods in Applied Mechanics and Engineering, vol.254, pp.197-221, 2013.

M. Signorini, S. Zlotnik, and P. Díez, Proper generalized decomposition solution of the parameterized Helmholtz problem: application to inverse geophysical problems, International Journal for Numerical Methods in Engineering, vol.109, issue.8, pp.1085-1102, 2017.

H. P. Thai, L. Chamoin, and C. Ha-minh, Robust a posteriori error estimation in isogeometric analysis using the concept of constitutive relation error, Computer Methods in Applied Mechanics and Engineering, 2018.

?. Temizer, P. Wriggers, and T. J. Hughes, Contact treatment in isogeometric analysis with NURBS, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.9, pp.1100-1112, 2011.

R. Verfürth, A review of a posteriori error estimation adaptive meshrefinement techniques, Journal of Computational and Applied Mathematics, vol.50, pp.67-83, 1994.

A. V. Vuong, C. Giannelli, B. Jüttler, and B. Simeon, A hierarchical approach to adaptive local refinement in isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.49, pp.3554-3567, 2011.

J. Waeytens, P. Chamoin, and L. Ladevéze, Guaranteed error bounds on pointwise quantities of interest for transient viscodynamics problems, Computational Mechanics, vol.49, issue.3, pp.291-307, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01580631

L. Wang, L. Chamoin, P. Ladevèze, and H. Zhong, Computable upper and lower bounds on eigenfrequencies, Computer Methods in Applied Mechanics and Engineering, vol.302, pp.27-43, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01241742

W. A. Wall, M. A. Frenzel, and C. Cyron, Isogeometric structural shape optimization, Computer Methods in Applied Mechanics and Engineering, vol.197, issue.33, pp.2976-2988, 2008.

C. Wang, S. Xia, X. Wang, and X. Qian, Isogeometric shape optimization on triangulations, Computer Methods in Applied Mechanics and Engineering, vol.331, pp.585-622, 2018.

G. Xu, B. Mourrain, R. Duvigneau, and A. Galligo, Parametrization of computational domain in isogeometric analysis: methods and comparison, Computer Methods in Applied Mechanics and Engineering, pp.2021-2031, 2011.
URL : https://hal.archives-ouvertes.fr/inria-00530758

G. Xu, B. Mourrain, R. Duvigneau, and A. Galligo, A New Error Assessment Method in Isogeometric Analysis of 2d Heat Conduction Problems
URL : https://hal.archives-ouvertes.fr/hal-00742955

, Advanced Science Letters, vol.10, issue.1, pp.508-512, 2012.

X. Zou, M. Conti, P. Díez, and F. Auricchio, A nonintrusive proper generalized decomposition scheme with application in biomechanics, International Journal for Numerical Methods in Engineering, vol.113, issue.2, pp.230-251, 2018.

S. Zlotnik, P. Díez, D. Modesto, and A. Huerta, Proper generalized decomposition of a geometrically parametrized heat problem with geoEngineering, vol.103, pp.737-758, 2015.

S. Zhu, L. Dedè, and A. Quarteroni, Isogeometric analysis and proper orthogonal decomposition for the acoustic wave equation, ESAIM: Mathematical Modelling and Numerical Analysis, vol.51, issue.4, pp.1197-1221, 2017.

O. C. Zienkiewicz, Displacement and equilibrium models in the finite element method by B. Fraeijs de Veubeke, International Journal for Numerical Methods in Engineering, vol.52, issue.3, pp.287-342, 1965.