. .. Introduction, 30 2.2 3D thermal problem within the Finite Element framework

C. .. Discussion,

. , 42 3.2.2 Progressive construction of the separated representation, p.42

. , Solution with an incremental scheme with FE framework

. .. , 47 3.4.1 PGD for linear 3D thermal problem under cyclic loading

. .. , PGD for nonlinear 3D thermal problem under cyclic loading, p.53

. .. Conclusion,

. , 3.1.2 Strategy to generate the time bases

. , 3.2.5 Non-constant source and non-homogeneous Dirichlet with different cycle times

. , Effect of the time basis order

, Usage of the time basis to generate solution with larger number of cycles, vol.113

. , Influence of the load amplitude

, Discussion on the use of the mixed strategy for linear problem, p.135

, Discussion on the use of the mixed strategy for nonlinear problem, p.145

. .. Conclusion, 145 1. la méthode de Newton-Raphson

. La-méthode-de-picard, , 2013.

. Chinesta, , 2013.

. La-méthode-d-;-aguado, , 2013.

, La dernière méthode est plus efficace dans la mesure où la non linéarité est évaluée sur une base plus réduite. L'évaluation de non linéarité lorsqu'une représentation séparée est utilisée reste à ce jour un thème exploré, ce n'est néanmoins pas l'objet de cette thèse où la méthode DEIM sera utilisée

. Aghighi, Elle consiste à transformer la fonction en une représentation dans le un grand nombre d'articles souligne les limitations de cette approche notamment pour l'étude de la fatigue des polymères : temps de calcul important, non-convergence et besoin d'une grande capacité mémoire. Pour surmonter ces limitations, les méthodes de réduction du modèles semblent être un moyen efficace et plus particulièrement la méthode PGD. Bibliography ***, L'analyse de Fourier sera appliquée dans la troisième approche étudiée dans cette thèse pour l'étude d'une fonction temporelle, vol.200, pp.65-78, 2013.

J. Aguado, Advanced strategies for the separated formulation of problems in the Proper Generalized Decomposition framework, 2015.
URL : https://hal.archives-ouvertes.fr/tel-01926078

[. Aguado, Deim-based pgd for parametric nonlinear model order reduction, Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), pp.29-34, 2013.

[. Al-takash, Utilisation d'une base temporelle a priori issue d'une analyse tfr pour accélérer la prédiction du comportement d'un polymère sous chargement cyclique, Congrès Français de Mécanique, 2017.

[. Al-takash, Numerical approach based on the collection of the most significant modes to solve cyclic transient thermal problems involving different time scales, Journal of Computational Physics, 2018.

[. Al-takash, On the validation of the proper generalized decomposition method with finite element method: 3d heat problem under cyclic loading, Mechanism, Machine, Robotics and Mechatronics Sciences, pp.3-13, 2019.

. Ammar, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids, Journal of Non-Newtonian Fluid Mechanics, vol.139, issue.3, pp.153-176, 2006.
URL : https://hal.archives-ouvertes.fr/hal-01004909

. Ammar, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids: Part ii: Transient simulation using space-time separated representations, Journal of Non-Newtonian Fluid Mechanics, vol.144, issue.2, pp.98-121, 2007.
URL : https://hal.archives-ouvertes.fr/hal-01633241

. Ammar, Non-Incremental Strategies Based on Separated Representations : Applications in Computational Rheology, Communications in Mathematical Sciences, vol.8, issue.3, pp.671-695, 2010.
URL : https://hal.archives-ouvertes.fr/hal-01007108

. Ammar, On the spacetime separated representation of integral linear viscoelastic models, Comptes Rendus Mécanique, vol.343, issue.4, pp.247-263, 2015.

T. L. Anderson, Fracture mechanics-fundamentals and applications, 1991.

T. L. Anderson, Fracture mechanics-fundamentals and applications, 2017.

D. Annaratone-;-annaratone, Transient Heat Transfer. SpringerBriefs in Applied Sciences and Technology, 2011.

. Ascher, Implicit-explicit rungekutta methods for time-dependent partial differential equations, Applied Numerical Mathematics, vol.25, issue.2, pp.151-167, 1997.

[. Barrault, , 2004.

, An 'empirical interpolation' method: application to efficient reduced-basis discretization of partial differential equations, Comptes Rendus Mathematique, vol.339, issue.9, pp.667-672

A. Benaarbia and A. Et-chrysochoos, Proper orthogonal decomposition preprocessing of infrared images to rapidly assess stress-induced heat source fields, Quantitative InfraRed Thermography Journal, vol.14, issue.1, pp.132-152, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01505316

. Benjamin, Reduced-order modelling for linear heat conduction with parametrised moving heat sources, vol.39, pp.170-188

[. Benner, A survey of projectionbased model reduction methods for parametric dynamical systems, SIAM review, vol.57, issue.4, pp.483-531, 2015.

. Bergheau, J. Bergheau, and R. Et-fortunier, Finite Element Simulation of Heat Transfer, 2010.

[. Bergheau, The proper generalized decomposition as a space-time integrator for elastoplastic problems, Comptes Rendus Mécanique, vol.344, issue.11, pp.759-768, 2016.

[. Beringhier, Solution of strongly coupled multiphysics problems using space-time separated representations-application to thermoviscoelasticity, Archives of Computational Methods in Engineering, vol.17, issue.4, pp.393-401, 2010.

A. Berrehili, Comportement cyclique et tenue en fatigue sous chargement multiaxial d'un polyéthylène : expériences et critère d'endurance, 2010.

M. Bhattacharyya, A model reduction approach in space and time for fatigue damage simulation, 2018.
URL : https://hal.archives-ouvertes.fr/tel-01808371

[. Bhattacharyya, New latin-pgd technique for fatigue loading, European Congress on Computational Methods in Applied Sciences and Engineering, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01735654

M. A. Bhatti, Fundamental finite element analysis and applications. Hoboken, 2005.

[. Bia?ecki, Proper orthogonal decomposition and modal analysis for acceleration of transient fem thermal analysis, International journal for numerical methods in engineering, vol.62, issue.6, pp.774-797, 2005.

[. Bognet, Advanced simulation of models defined in plate geometries: 3d solutions with 2d computational complexity, Computer Methods in Applied Mechanics and Engineering, pp.201-204, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01462825

[. Boisse, A new approach in non-linear mechanics: The large time increment method, International Journal for Numerical Methods in Engineering, vol.29, issue.3, pp.647-663, 1990.

L. Brandolini, Fourier analysis and convexity, 2004.

[. Capaldo, An approximation framework dedicated to PGD-based nonlinear solver, 11th World Congress on Computational Mechanics (WCCM XI), 2014.
URL : https://hal.archives-ouvertes.fr/hal-01694120

. Carslaw, H. Jaeger-;-carslaw, and J. Jaeger, Conduction of Heat in Solids, Oxford science publications, 1986.

[. Chama, Newton raphson solver for finite element methods featuring nonlinear hysteresis models, IEEE Transactions on Magnetics, vol.54, issue.1, pp.1-8, 2018.

A. Chatterjee, An introduction to the proper orthogonal decomposition, Current science, vol.78, pp.808-817, 2000.

S. Chaturantabut and D. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM Journal on Scientific Computing, vol.32, issue.5, pp.2737-2764, 2010.

[. Chinesta, On the use of proper generalized decompositions for solving the multidimensional chemical master equation, European Journal of Computational Mechanics/Revue Européenne de Mécanique Numérique, vol.19, issue.1-3, pp.53-64, 2010.
URL : https://hal.archives-ouvertes.fr/hal-01007161

[. Chinesta, 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

[. Chinesta, Proper generalized decomposition of multiscale models, International Journal for Numerical Methods in Engineering, vol.83, pp.1114-1132, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01007223

[. Chinesta, On the reduction of stochastic kinetic theory models of complex fluids. Modelling and Simulation in, Materials Science and Engineering, vol.15, issue.6, pp.639-652, 2007.
URL : https://hal.archives-ouvertes.fr/hal-01476037

[. Chinesta, An overview of the proper generalized decomposition with applications in computational rheology, Journal of Non-Newtonian Fluid Mechanics, vol.166, issue.11, pp.578-592, 2011.
URL : https://hal.archives-ouvertes.fr/hal-01061441

[. Chinesta, The proper generalized decomposition for advanced numerical simulations: a primer, 2013.

F. Chinesta-et-ladevèze-;-chinesta and P. Et-ladevèze, Separated Representations and PGD-Based Model Reduction, 2014.

[. Chinesta, A short review on 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

[. Chinesta, Towards a Framework for Non-Linear Thermal Models in Shell Domains, International Journal of Numerical Methods for Heat and Fluid Flow, vol.23, issue.1, pp.55-73, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01007391

[. Chinesta, 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

. Cochelin, The asymptotic-numerical method: an efficient perturbation technique for nonlinear structural mechanics, vol.3, pp.281-297, 1994.

. Cochelin, Asymptotic-numerical methods and pade approximants for non-linear elastic structures, International Journal for Numerical Methods in Engineering, vol.37, issue.7, pp.1187-1213, 1994.

J. Cognard and P. Et-ladevèze, A large time increment approach for cyclic viscoplasticity, International Journal of plasticity, vol.9, issue.2, pp.141-157, 1993.

[. Comte, A direct method for the solution of evolution problems, Comptes Rendus Mécanique, vol.334, issue.5, pp.317-322, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00830685

[. Constantinescu, A computational approach to thermomechanical fatigue, International Journal of fatigue, vol.26, issue.8, pp.805-818, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00022603

J. W. Cooley, The re-discovery of the fast fourier transform algorithm, Microchimica Acta, vol.93, issue.1, pp.33-45, 1987.

E. Cueto and F. Chinesta, Real time simulation for computational surgery: a review, Advanced Modeling and Simulation in Engineering Sciences, vol.1, pp.1-11, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01590981

[. Cueto, Proper Generalized Decomposition: An Introduction to Computer Implementation with Matlab, SpringerBriefs in Applied Sciences and Technology, 2016.

N. Potier-ferry-;-damil and M. Potier-ferry, A new method to compute perturbed bifurcations: Application to the buckling of imperfect elastic structures, International Journal of Engineering Science, vol.28, issue.9, pp.943-957, 1990.

M. Dokainish and K. Subbaraj, A survey of direct timeintegration methods in computational structural dynamics. explicit methods, Computers & Structures, vol.32, issue.6, pp.1371-1386, 1989.

. Santos, Thermal conductivity behaviour of polymers around glass transition and crystalline melting temperatures, Polymer Testing, vol.32, issue.5, pp.987-994, 2013.

M. O. Efe and H. Ozbay, Proper orthogonal decomposition for reduced order modeling: 2d heat flow, Control Applications, 2003. CCA 2003. Proceedings of 2003 IEEE Conference on, vol.2, pp.1273-1277, 2003.

D. Eyre-et-milton-;-eyre and G. Milton, A fast numerical scheme for computing the response of composites using grid refinement, The European Physical Journal Applied Physics, vol.6, issue.1, pp.41-47, 1999.

B. Feeny and R. Kappagantu, On the physical interpretation of proper orthogonal modes in vibrations, Journal of sound and vibration, vol.211, issue.4, pp.607-616, 1998.

[. Felder, Multiscale fe-fft-based thermo-mechanically coupled modeling of viscoplastic polycrystalline materials, 7th GACM Colloquium on Computational Mechanics for Young Scientists from Academia and Industry, 2017.

[. Fritzen, An algorithmic comparison of the hyper-reduction and the discrete empirical interpolation method for a nonlinear thermal problem, Mathematical and Computational Applications, vol.23, pp.8-31, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01759176

[. Fu, Pod/deim reduced-order modeling of time-fractional partial differential equations with applications in parameter identification, Journal of Scientific Computing, vol.74, issue.1, pp.220-243, 2018.

[. Ghavamian, Pod-deim model order reduction for strain-softening viscoplasticity, Computer Methods in Applied Mechanics and Engineering, vol.317, pp.458-479, 2017.

[. Ghnatios, On the space separated representation when addressing the solution of pde in complex domains, Discrete and Continuous Dynamical Systems-S, vol.9, pp.475-500, 2016.

[. Giner, The proper generalized decomposition (pgd) as a numerical procedure to solve 3d cracked plates in linear elastic fracture mechanics, International Journal of Solids and Structures, vol.50, issue.10, pp.1710-1720, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01007373

M. A. Grepl, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations, ESAIM: M2AN, vol.41, issue.3, pp.575-605, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00112154

[. Guérin, Thermomechanical model reduction for efficient simulations of rotor-stator contact interaction, ASME Turbo Expo 2018: Turbomachinery Technical Conference and Exposition, 2018.

[. Hammoud, A reduced simulation applied to the viscoelastic fatigue of polymers, Comptes Rendus Mécanique, vol.342, issue.12, pp.671-691, 2014.

[. Har?arik, Frequency analysis of acoustic signal using the fast fourier transformation in matlab, Modelling of Mechanical and Mechatronics Systems, vol.48, pp.199-204, 2012.

P. Holmes, Turbulence, coherent structures, dynamical systems and symmetry, 2012.

. Hughes, . Liu, T. J. Hughes, and W. K. Liu, Implicit-explicit finite elements in transient analysis stability theory, Journal of Applied Mechanics, vol.45, pp.371-374, 1978.

[. Ichihashi, Proper orthogonal decomposition and fourier analysis on the energy release rate dynamics of a gas turbine combustor, 48th AIAA Aerospace Science Meeting, 2010.

A. Khennane, Introduction to finite element analysis using MATLAB® and abaqus, 2013.

P. Ladevèze, Sur une famille d'algortithmes en mécanique de structure, Comptes-rendus de l'Académie des Sciences, pp.41-44, 1985.

P. Ladeveze and A. Nouy, On a multiscale computational strategy with time and space homogenization for structural mechanics, Computer Methods in Applied Mechanics and Engineering, vol.192, issue.28, pp.3061-3087, 2003.
URL : https://hal.archives-ouvertes.fr/hal-00366648

[. Ladeveze, The latin multiscale computational method and the proper generalized decomposition, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.21, pp.1287-1296, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00414447

M. Larson and F. Et-bengzon, The Finite Element Method: Theory, Implementation, and Applications. Mathematics and statics, 2013.

L. Bris-;-le-bris and C. , Mathematical and numerical analysis for molecular simulation: Accomplishments and challenges, Proceedings of the International Congress of Mathematicians, vol.3, pp.1507-1522, 2006.

[. Lebensohn, Fast fourier transform-based modeling for the determination of micromechanical fields in polycrystals, vol.63, pp.13-18, 2011.

[. Lebensohn, An elasto-viscoplastic formulation based on fast fourier transforms for the prediction of micromechanical fields in polycrystalline materials, International Journal of Plasticity, pp.59-69, 2012.

. Liang, Proper orthogonal decomposition and its applications-part i: Theory, Journal of Sound and vibration, vol.252, issue.3, pp.527-544, 2002.

. Lihong, Adaptive pod-deim basis construction and its application to a nonlinear population balance system, AIChE Journal, vol.63, issue.9, pp.3832-3844, 2017.

[. Maday, A general multipurpose interpolation procedure: the magic points, Communications on Pure and Applied Analysis, vol.8, pp.383-404, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00174797

E. Marin-;-marin, Characteristic dimensions for heat transfer, Latin-American Journal of Physics Educationl, vol.4, pp.56-60, 2010.

. Miled, A priori hyperreduction method for coupled viscoelastic-viscoplastic composites, Computers & Structures, vol.119, pp.95-103, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00790175

C. Montebello, Analysis of the stress gradient effect in FrettingFatigue through a description based on nonlocal intensity factors, 2015.
URL : https://hal.archives-ouvertes.fr/tel-01238905

, Dimensional reduction for the simulation of metal fatigue, 2017.

[. Nasri, Proper generalized decomposition (pgd) for the numerical simulation of polycrystalline aggregates under cyclic loading, Comptes Rendus Mécanique, vol.346, issue.2, pp.132-151, 2018.

A. Nayebi and M. Hamidpour, Thermo-mechanical cyclic loading analysis of pipes with different type of defects: Temperature dependent properties. Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, vol.230, issue.1, pp.303-310, 2016.

D. Néron and P. Et-ladevèze, Proper generalized decomposition for multiscale and multiphysics problems, Archives of Computational Methods in Engineering, vol.17, issue.4, pp.351-372, 2010.

. Néron, Accounting for nonlinear aspects in multiphysics problems: Application to poroelasticity, Lecture notes in computer science, vol.3039, pp.612-620, 2004.

A. J. Newman, Model reduction via the karhunen-loeve expansion part i: An exposition, 1996.

S. T. Nguyen, Caractérisation expérimentale et modélisation thermomécanique de l'accommodation cyclique du polyéthylene, 2013.

. Nguyen, Nonlinear viscoelastic contribution to the cyclic accommodation of high density polyethylene in tension: Experiments and modeling, International journal of fatigue, vol.55, pp.166-177, 2013.

T. L. Nguyen, La Décomposition propre généralisée pour la résolution de problèmes multiphysiques transitoires couplés dédiés à la mécanique des matériaux, 2012.

[. Niroomandi, Model order reduction for hyperelastic materials, International Journal for Numerical Methods in Engineering, vol.81, issue.9, pp.1180-1206, 2009.
URL : https://hal.archives-ouvertes.fr/hal-01007059

B. R. Noack, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2013.

. Noor, A. K. Peters-;-noor, and J. M. Peters, Reduced basis technique for nonlinear analysis of structures, AIAA Journal, vol.18, issue.4, pp.455-462, 1980.

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

M. Othman and I. Abbas, Fundamental solution of generalized thermo-viscoelasticity using the finite element method, Computational Mathematics and Modeling, vol.23, pp.158-167, 2012.

M. Peigney and C. Stolz, An optimal control approach to the analysis of inelastic structures under cyclic loading, Journal of the Mechanics and Physics of Solids, vol.51, issue.4, pp.575-605, 2003.

[. Prulière, 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.

[. Prulière, An efficient reduced simulation of residual stresses in composite forming processes, International Journal of Material Forming, vol.3, issue.2, pp.1339-1350, 2010.

S. S. Rao and J. N. Reddy, An introduction to the finite element method, vol.2, 1993.

R. Rizk and M. Et-awad, Mechanism, Machine, Robotics and Mechatronics Sciences. Mechanisms and Machine Science, Comptes Rendus Mécanique, vol.330, issue.7, pp.499-505, 2002.

D. Ryckelynck, A priori hyperreduction method: an adaptive approach, Journal of computational physics, vol.202, issue.1, pp.346-366, 2005.

D. Ryckelynck and D. M. Et-benziane, Multi-level a priori hyper-reduction of mechanical models involving internal variables, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.17, pp.1134-1142, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00461492

[. Ryckelynck, On the a priori model reduction: overview and recent developments, Archives of Computational methods in Engineering, vol.13, issue.1, pp.91-128, 2006.
URL : https://hal.archives-ouvertes.fr/hal-01007164

[. Ryckelynck, Estimation of the validity domain of hyper-reduction approximations in generalized standard elastoviscoplasticity. Advanced Modeling and Simulation in Engineering Sciences, vol.2, pp.6-25, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01237733

J. A. Sauer-et-richardson-;-sauer and G. C. Et-richardson, Fatigue of polymers, International Journal of Fracture, vol.16, issue.6, pp.499-532, 1980.

K. Spiliopoulos, A simplified method to predict the steady cyclic stress state of creeping structures, Journal of applied mechanics, vol.69, issue.2, pp.148-153, 2002.

M. Stehly and Y. Et-remond, On numerical simulation of cyclic viscoplastic and viscoelastic constitutive laws with the large time increment method, Mechanics of Time-Dependent Materials, vol.6, issue.2, pp.147-170, 2002.

[. Stein, Encyclopedia of computational mechanics, vol.1, 2004.

K. Subbaraj and M. Dokainish, A survey of direct timeintegration methods in computational structural dynamics. implicit methods, Computers & Structures, vol.32, issue.6, pp.1387-1401, 1989.

H. J. Sutherland, Frequency domain analysis of the fatigue loads on typical wind turbine blades, Journal of Solar Engineering, p.118, 1996.

T. Szabó-;-szabó, On the discretization time-step in the finite element theta-method of the discrete heat equation, Numerical Analysis and Its Applications, pp.564-571, 2009.

C. Van-loan, Computational Frameworks for the Fast Fourier Transform, Society for Industrial and Applied Mathematics, 1992.

D. F. Walnut, An introduction to wavelet analysis, 2013.

. Zienkiewicz, O. C. Taylor-;-zienkiewicz, and R. L. Taylor, The finite element method: solid mechanics, vol.2, 2000.

. Zienkiewicz, O. C. Taylor-;-zienkiewicz, and R. L. Taylor, The finite element method for solid and structural mechanics, vol.6, 2005.

, Computing S(t) from R(x) and W (k) The virtual field is written as: u * (x, t, k) = R(x)