V. Polymer-viscoelasticity, 74 3.2.1 The Pom-Pom model and why? 76 Model simplifications, advantages, disadvantages and solutions, p.77

.. Isothermal-viscoelasticity, Short review on computational methods used in viscoelastic flow, p.79

.. Mixed, Elastic Viscous Stress Splitting), 80 D.E.V.S.S. (Discrete Elastic Viscous Stress Splitting) . . . . . . . . . . . . . . . 80

.. Some-remarks-on-thermal-viscoelasticity, 107 3.6.1 Thermodynamic point of view of non-isothermal viscoelasticity and state of the art, p.107

.. Temperature-equation-for-viscoelastic-compressible-flows, 109 Formulation of the Pom-Pom model with a conformation tensor 109 Temperature equation of a one-mode fluid, p.110

E. Results and .. Comparison-with-simulation, 122 4.3.1 Molds and equipment, p.124

R. Comparison, 132 Experimental and numerical short-shots, p.133

.. Pressure-trace, 138 4.4.3 Mass and shrinkage rate, Orientation and stretch distributions . . . . . . . . . . . . . . . . . . . . . . . . 139

. Arnold, A stable finite element for Stokes equations. Calcolo, pp.337-344, 1984.

F. Baaijens, Calculation of residual stresses in injection molded products, Rheologica Acta, vol.30, issue.3, pp.284-299, 1991.
DOI : 10.1007/BF00366642

F. Baaijens, Mixed finite element methods for viscoelastic flow analysis: a review, Journal of Non-Newtonian Fluid Mechanics, vol.79, issue.2-3, pp.361-386, 1998.
DOI : 10.1016/S0377-0257(98)00122-0

. Baaijens, Viscoelastic flow past a confined cylinder of a low density polyethylene melt, Journal of Non-Newtonian Fluid Mechanics, vol.68, issue.2-3, pp.173-203, 1998.
DOI : 10.1016/S0377-0257(96)01519-4

. Batkam, A Space-Time Discontinuous Galerkin Method for Convection and Diffusion in Injection Moulding, International Journal of Forming Processes, vol.7, issue.1-2, 2003.
DOI : 10.3166/ijfp.7.11-33

URL : https://hal.archives-ouvertes.fr/hal-00614585

. Beraudo, A finite element method for computing the flow of multi-mode viscoelastic fluids: comparison with experiments, Journal of Non-Newtonian Fluid Mechanics, vol.75, issue.1, pp.1-23, 1998.
DOI : 10.1016/S0377-0257(97)00083-9

A. Beris and B. Edwards, Thermodynamic of flowing systems, 1994.

E. Bigot and T. Coupez, Capture of 3d moving free surfaces and material interfaces by mesh deformation, ECCOMAS, pp.11-14, 2000.

. Bikard, Polyurethane foam expansion macro-modelling, International Journal for Numerical Methods in Fluids, 2004.

. Blackwell, Molecular drag???strain coupling in branched polymer melts, Journal of Rheology, vol.44, issue.1, pp.121-136, 2000.
DOI : 10.1122/1.551081

. Bogaerds, 3D Viscoelastic analysis of a polymer solution in a complex flow, Computer Methods in Applied Mechanics and Engineering, vol.180, issue.3-4, pp.413-430, 1999.
DOI : 10.1016/S0045-7825(99)00176-0

. Bourrigaud, Shear Modification of Long-Chain Branched Polymers:?? A Theoretical Approach Using the Pom-Pom Model, Macromolecules, vol.36, issue.4, pp.1388-1394, 2003.
DOI : 10.1021/ma021195t

. Brezzi, . Fortin, F. Brezzi, and M. Fortin, Mixed and hybrid finite element methods, 1991.
DOI : 10.1007/978-1-4612-3172-1

. Bristeau, Compressible viscous flow calculations using compatible finite element approximations, International Journal for Numerical Methods in Fluids, vol.18, issue.6, pp.719-749, 1990.
DOI : 10.1002/fld.1650110603

. Broyer, Flow-analysis network (fan) method for solving flow problems in polymer processing, Polymer Engineering and Science, vol.14, p.660, 1974.

J. Bruchon and T. Coupez, The foam structure prediction by the 3d calculation of the bubble growth in a molten polymer, Journal of Non-Newtonian Fluid Mechanics, 2003.

E. Brujan, The equation of bubble dynamics in a compressible linear viscoelastic liquid, Fluid Dynamics Research, vol.29, issue.5, pp.287-294, 2001.
DOI : 10.1016/S0169-5983(01)00030-2

M. Chang, On the study of surface defects in the injection moulding of rubbermodified thermoplastics, pp.360-367, 1994.

. Chiang, A unified simulation of filling and post-filling stages in injection molding. part 1: formulation, Polymer Engineering and Science, issue.2, pp.31116-140, 1991.

. Chinesta, A semi-lagrangian strategy to predict the fiber orientation in the steady flows of reinforced thermoplastics, Computer Methods in Applied Mechanics and Engineering, vol.189, issue.1, pp.233-247, 2000.
DOI : 10.1016/S0045-7825(99)00325-4

. Clemeur, On the evaluation of some differential formulations for the 'pom-pom' constitutive model, Rheologica Acta, vol.14, issue.1, pp.25-32, 2002.

. Cormenzana, Calculation of free surface flows using CONNFFESSIT, Journal of Rheology, vol.45, issue.1, pp.237-258, 2001.
DOI : 10.1122/1.1333000

T. Coupez, Stable-stabilized finite element for 3d forming calculation, 1996.

R. Keunings, Finite element analysis of die swell of a highly elastic fluid, Journal of Non-Newtonian Fluid Mechanics, vol.10, pp.339-356, 1982.

M. Cruickshank, J. Cruickshank, and B. Munson, Viscous buckling of plane and axisymetric jets, Journal of Fluid Mechanics, vol.13, pp.221-239, 1981.

G. D. Davis, Natural convection of air in a square cavity: A bench mark numerical solution, International Journal for Numerical Methods in Fluids, vol.12, issue.3, pp.249-264, 1983.
DOI : 10.1002/fld.1650030305

W. Dee, G. Dee, and D. Walsh, Equations of state for polymer liquids, Macromolecules, vol.21, issue.3, pp.815-817, 1988.
DOI : 10.1021/ma00181a043

J. Des-cloizeaux-]-des-cloizeaux, Relaxation of entangled polymers in melts, Macromolecules, vol.23, issue.17, pp.3992-4006, 1989.
DOI : 10.1021/ma00219a021

. Dietz, Orientation development and relaxation in injection molding of amorphous polymers, Polymer Engineering and Science, vol.50, issue.4, p.273, 1978.
DOI : 10.1002/pen.760180407

H. Digonnet and T. Coupez, Object-oriented programming for ???fast and easy??? development of parallel applications in forming processes simulation, Second MIT Conference on Computational Fluid and Solid Mechanics Massachussets Institute of Technology, pp.1922-1924, 2003.
DOI : 10.1016/B978-008044046-0.50471-1

URL : https://hal.archives-ouvertes.fr/hal-00531577

M. Doi and S. Edwards, The theory of polymer dynamics, 1986.

. Dressler, Macroscopic thermodynamics of flowing polymeric liquids, Rheologica Acta, vol.38, issue.2, pp.117-136, 1999.
DOI : 10.1007/s003970050162

A. Flaman, Built-up and relaxation of molecular orientation in injection molding, 1990.

R. Fulchiron, Comportement des matériaux sous-pression, 2002.

P. D. Gennes, Reptation of a Polymer Chain in the Presence of Fixed Obstacles, The Journal of Chemical Physics, vol.55, issue.2, pp.572-579, 1970.
DOI : 10.1063/1.1675789

. Graham, Using the pom-pom equations to analyze polymer melts in exponential shear, Journal of Rheology, vol.45, issue.1, pp.27-52, 2001.
DOI : 10.1122/1.1332999

. Greener, . Pearson, J. Greener, and G. Pearson, Orientation Residual Stresses and Birefringence in Injection Molding, Journal of Rheology, vol.27, issue.2, p.115, 1983.
DOI : 10.1122/1.549701

F. Guenette, R. Guenette, and M. Fortin, A new mixed finite element method for computing viscoelastic flows, Journal of Non-Newtonian Fluid Mechanics, vol.60, issue.1, pp.275-290, 1995.
DOI : 10.1016/0377-0257(95)01372-3

C. Hammer, Calcul des contraintes et deformations residuelles lors de la mise en oeuvre de verre ophtalmique en polymère thermodurcissable, 2001.

A. Harlow, F. Harlow, and A. Amsden, A numerical fluid dynamics calculation method for all flow speeds, Journal of Computational Physics, vol.8, issue.2, pp.197-213, 1971.
DOI : 10.1016/0021-9991(71)90002-7

P. Harry, D. Harry, and R. Parrot, Numerical simulation of injection mold filling, Polymer Engineering and Science, vol.14, issue.4, p.209, 1970.
DOI : 10.1002/pen.760100404

H. Hauke, G. Hauke, and T. Hughes, A unified approach to compressible and incompressible flows, Computer Methods in Applied Mechanics and Engineering, vol.113, issue.3-4, pp.389-395, 1994.
DOI : 10.1016/0045-7825(94)90055-8

S. Hieber, C. Hieber, and S. Shen, A finite-element/finite-difference simulation of the injection-molding filling process, Journal of Non-Newtonian Fluid Mechanics, vol.7, issue.1, pp.1-32, 1980.
DOI : 10.1016/0377-0257(80)85012-9

D. Hoff, Global Solutions of the Navier-Stokes Equations for Multidimensional Compressible Flow with Discontinuous Initial Data, Journal of Differential Equations, vol.120, issue.1, pp.215-254, 1995.
DOI : 10.1006/jdeq.1995.1111

H. Illinca, F. Illinca, and J. Hetu, Three-dimensional Filling and Post-filling Simulation of Polymer Injection Molding, International Polymer Processing, vol.16, issue.3, pp.291-301, 2001.
DOI : 10.3139/217.1643

. Inkson, Predicting low density polyethylene melt rheology in elongational and shear flows with ???pom-pom??? constitutive equations, Journal of Rheology, vol.43, issue.4, pp.873-869, 1999.
DOI : 10.1122/1.551036

A. Isayev, Orientation development in the injection molding of amorphous polymers, Polymer Engineering and Science, vol.8, issue.5, p.271, 1983.
DOI : 10.1002/pen.760230507

A. Isayev, Injection and Compression Molding Fundamentals, 1987.

H. Isayev, A. Isayev, and C. Hieber, Toward a viscoelastic modelling of the injection molding of polymers, Rheologica Acta, vol.15, issue.2, pp.168-182, 1980.
DOI : 10.1007/BF01521928

. Issa, The computation of compressible and incompressible recirculating flows by a non-iterative implicit scheme, Journal of Computational Physics, vol.62, issue.1, pp.66-82, 1986.
DOI : 10.1016/0021-9991(86)90100-2

H. Janeschitz, Injection moulding of plastics: Some ideas about the relationship between mould filling and birefringence, Rheologica Acta, vol.15, issue.4, p.327, 1977.
DOI : 10.1007/BF01534086

. Kamal, Computer simulation of injection mold filling for viscoelastic melts with fountain flow, Polymer Engineering and Science, vol.191, issue.3, pp.190-196, 1986.
DOI : 10.1002/pen.760260303

K. Kamal, M. Kamal, and S. Kenig, The injection molding of thermoplastics part I: Theoretical model, Polymer Engineering and Science, vol.66, issue.4, pp.294-308, 1972.
DOI : 10.1002/pen.760120408

Y. Kanel, A model system for the one-dimensional motion of a gas, Differencial'nya Uravenenija, pp.721-734, 1968.

L. Kellog, R. Kellog, and B. Liu, A Finite Element Method for the Compressible Stokes Equations, SIAM Journal on Numerical Analysis, vol.33, issue.2, pp.780-788, 1996.
DOI : 10.1137/0733039

L. Kellog, R. Kellog, and B. Liu, A Penalized Finite-Element Method for a Compressible Stokes System, SIAM Journal on Numerical Analysis, vol.34, issue.3, pp.1093-1105, 1997.
DOI : 10.1137/S0036142994273276

. Keshtiban, Numerical simulation of compressible viscoelastic liquids, Journal of Non-Newtonian Fluid Mechanics, vol.122, issue.1-3, pp.131-146, 2004.
DOI : 10.1016/j.jnnfm.2003.12.008

R. Keunings, An algorithm for the simulation of transient viscoelastic flows with free surfaces, Journal of Computational Physics, vol.62, issue.1, pp.199-220, 1986.
DOI : 10.1016/0021-9991(86)90107-5

B. Keunings, R. Keunings, and D. Bousfield, Analysis of surface tension driven leveling in viscoelastic films, Journal of Non-Newtonian Fluid Mechanics, vol.22, issue.2, pp.219-233, 1987.
DOI : 10.1016/0377-0257(87)80037-X

. Khazikov, . Shelukh, A. Khazikov, and V. Shelukh, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, Journal of Applied Mathematics and Mechanics, vol.41, issue.2, pp.273-282, 1977.
DOI : 10.1016/0021-8928(77)90011-9

. Khazikov, . Weigant, A. Khazikov, and V. Weigant, On the existence of global solutions of two-dimensional navier-stokes equations of a compressible viscous fluids, Journal of Siberian Mathematics, vol.36, pp.1108-1141, 1995.

. Kim, Numerical modeling of injection/compression molding for center-gated disk: Part I. Injection molding with viscoelastic compressible fluid model, Polymer Engineering & Science, vol.15, issue.10, pp.1930-1942, 1999.
DOI : 10.1002/pen.11586

. Kuo, . Kamal, Y. Kuo, and M. Kamal, The fluid mechanics and heat transfer of injection mold filling of thermoplastic materials, AIChE Journal, vol.22, issue.4, pp.661-669, 1976.
DOI : 10.1002/aic.690220405

. Kuo, . Kamal, Y. Kuo, and M. Kamal, Flow of thermoplastics in the filling and packing stages of injection molding, International Conference on Polymer Processing, pp.329-348, 1977.

Y. Kwon, On Hadamard stability for compressible viscoelastic constitutive equations, Journal of Non-Newtonian Fluid Mechanics, vol.65, issue.2-3, pp.151-163, 1996.
DOI : 10.1016/0377-0257(96)01457-7

. Lee, Experimental observation and numerical simulation of transient ???stress fangs??? within flowing molten polyethylene, Journal of Rheology, vol.45, issue.6, pp.451261-1277, 2001.
DOI : 10.1122/1.1389316

L. Lefebvre, Numerical simulation of the flow of polyurethane foams, 1993.

A. Leonov, Nonequilibrium thermodynamics and rheology of viscoelastic polymer media, Rheologica Acta, vol.14, issue.2, pp.85-98, 1976.
DOI : 10.1007/BF01517499

P. Lions, Existence globale de solutions pour les equations de navier-stokes compressibles isentropiques. Comptes-Rendus de l, Académie des Sciences, vol.316, pp.1335-1340, 1993.

H. Lord, Flow of polymers with pressure-dependent viscosity in injection molding dies, Polymer Engineering and Science, vol.20, issue.7, pp.469-473, 1979.
DOI : 10.1002/pen.760190702

M. Mahishi, Material characterization for thin wall molding simulation, ANTEC1998, pp.547-551, 1998.

I. Maillot, Simulation du remplissage/compactage pour la mise en forme des thermoplastiques par injection. Applications industrielles, 1993.

C. Marchal, J. Marchal, and M. Crochet, A new mixed finite element for calculating viscoelastic flow, Journal of Non-Newtonian Fluid Mechanics, vol.26, issue.1, pp.77-114, 1987.
DOI : 10.1016/0377-0257(87)85048-6

G. Marrucci, Testing of a constitutive equation for entangled networks by elongational and shear data of polymer melts, Rheologica Acta, vol.16, issue.2, p.269, 1983.
DOI : 10.1007/BF01635115

G. Marrucci, Relaxation by reptation and tube enlargement: A model for polydisperse polymers, Journal of Polymer Science: Polymer Physics Edition, vol.23, issue.1, pp.159-177, 1985.
DOI : 10.1002/pol.1985.180230115

N. Matsumura, A. Matsumura, and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proceedings of the Japan Academy, Series A, Mathematical Sciences, vol.55, issue.9, pp.337-342, 1979.
DOI : 10.3792/pjaa.55.337

. Matusu-necasova, Existence of classical solutions for compressible viscoelastic fluids of Oldroyd type past an obstacle, Mathematical Methods in the Applied Sciences, vol.320, issue.5, pp.449-460, 1999.
DOI : 10.1002/(SICI)1099-1476(19990325)22:5<449::AID-MMA34>3.0.CO;2-G

. Mavridis, Finite element simulation of fountain flow in injection molding, Polymer Engineering and Science, vol.50, issue.7, p.449, 1988.
DOI : 10.1002/pen.760260702

L. Mcleish, T. Mcleish, and R. Larson, Molecular constitutive equations for a class of branched polymers: The pom-pom polymer, Journal of Rheology, vol.42, issue.1, pp.81-110, 1998.
DOI : 10.1122/1.550933

. Nigro, A general algorithm for compressible and incompressible flow. Stability analysis and explicit time integration, International Journal of Numerical Methods for Heat & Fluid Flow, vol.7, issue.2/3, pp.141-168, 1997.
DOI : 10.1108/09615539710163239

N. Nonaka, N. Nonaka, and T. Nakayama, A unified method for the numerical analysis of compressible and incompressible viscous flows, Computational Mechanics, vol.35, issue.5, pp.369-376, 1996.
DOI : 10.1007/BF00376132

H. Ottinger, Thermodynamic admissibility of the pompon model for branched polymers, Rheologica Acta, vol.40, issue.4, pp.317-321, 2001.
DOI : 10.1007/s003970000159

. Pantani, Molecular orientation in injection molding: experiments and analysis, Rheologica Acta, vol.43, issue.2, pp.109-118, 2004.
DOI : 10.1007/s00397-003-0325-8

B. Peters, G. Peters, and F. Baaijens, Modelling of non-isothermal viscoelastic flows, Journal of Non-Newtonian Fluid Mechanics, vol.68, issue.2-3, pp.205-224, 1997.
DOI : 10.1016/S0377-0257(96)01511-X

C. Pichelin, E. Pichelin, and T. Coupez, Finite element solution of the 3D mold filling problem for viscous incompressible fluid, Computer Methods in Applied Mechanics and Engineering, vol.163, issue.1-4, pp.359-371, 1998.
DOI : 10.1016/S0045-7825(98)00024-3

URL : https://hal.archives-ouvertes.fr/hal-00617037

A. Pontes, Shrinkage and ejection forces in injection molded products, 2002.

. Rajagopalam, Finite element methdos for calculation of steady, viscoelastic flow using constitutive equations with a Newtonian viscosity, Journal of Non-Newtonian Fluid Mechanics, vol.36, pp.159-192, 1990.
DOI : 10.1016/0377-0257(90)85008-M

A. Redjeb, Simulation numérique de l'orientation de fibre en injection de thermoplastiques renforcés, 2003.

P. Rodgers, Pressure???volume???temperature relationships for polymeric liquids: A review of equations of state and their characteristic parameters for 56 polymers, Journal of Applied Polymer Science, vol.48, issue.6, pp.1061-1080, 1993.
DOI : 10.1002/app.1993.070480613

R. , C. Ryan, M. Chung, and T. , Conforming mapping analysis of injection mold filling, Polymer Engineering and Science, vol.20, p.642, 1980.

E. Saez, ´ Etude numérique du remplissage 3D en fonderie, 2003.

. Shyu, Numerical simulation of flow induced birefringence in injection molded disk, Korea-Australia Rheology Journal, vol.15, pp.159-166, 2003.

S. Simha, R. Simha, and T. Somcynski, On the Statistical Thermodynamics of Spherical and Chain Molecule Fluids, Macromolecules, vol.2, issue.4, pp.342-350, 1969.
DOI : 10.1021/ma60010a005

J. Simo, On a fully three-dimensional finite-strain viscoelastic damage model: Formulation and computational aspects, Computer Methods in Applied Mechanics and Engineering, vol.60, issue.2, pp.153-173, 1987.
DOI : 10.1016/0045-7825(87)90107-1

I. Sirakov, Etude parélémentsparéléments finis desécoulementsdesécoulements viscoélastiques des polymères fondus dans des géometries complexes. Résultats numériques et expérimentaux, 2000.

. Smirnova, Structure Development in Injection Molding: A 3D Simulation with a Differential Formulation of the Kinetic Equations, International Polymer Processing, vol.20, issue.2, 2004.
DOI : 10.3139/217.1873

. Sun, An adaptive viscoelastic stress splitting scheme and its applications: AVSS/SI and AVSS/SUPG, Journal of Non-Newtonian Fluid Mechanics, vol.65, issue.1, pp.75-91, 1996.
DOI : 10.1016/0377-0257(96)01448-6

. Szadi, A new mixed finite element method for viscoelastic flows governed by differential constitutive equations, Journal of Non-Newtonian Fluid Mechanics, vol.59, issue.2-3, pp.215-243, 1995.
DOI : 10.1016/0377-0257(95)01370-B

F. Takeshima, M. Takeshima, and N. Funakoshi, Molecular orientation distribution in injection-molded polycarbonate discs, Journal of Applied Polymer Science, vol.32, issue.2, pp.3457-3468, 2001.
DOI : 10.1002/app.1986.070320209

R. Tanner, A theory of die-swell, Journal of Polymer Science Part A-2: Polymer Physics, vol.8, issue.12, pp.2067-2078, 1970.
DOI : 10.1002/pol.1970.160081203

. Titomanlio, Modelling of the filling and packing step in the injection molding of thermoplastic materials, Ing.Chim.It, vol.16, p.110, 1980.

R. Valette, The matching of numerical simulation involving time dependent compressible viscoelastic flow with precise polyethylene processing data, 2003.

. Verbeeten, Viscoelastic analysis of complex polymer melt flows using the eXtended Pom???Pom model, Journal of Non-Newtonian Fluid Mechanics, vol.108, issue.1-3, pp.301-326, 2002.
DOI : 10.1016/S0377-0257(02)00136-2

S. Wakashima, S. Wakashima, and T. Saitoh, Benchmark solutions for natural convection in a cubic cavity using the high-order time???space method, International Journal of Heat and Mass Transfer, vol.47, issue.4, pp.853-864, 2004.
DOI : 10.1016/j.ijheatmasstransfer.2003.08.008

J. Wales, The application of flow birefringence to rheological studies of polymer melts, 1976.
DOI : 10.1007/978-94-010-1556-1

. Wang, Summary, Journal of Polymer Engineering, vol.7, issue.1, pp.21-45, 1986.
DOI : 10.1515/polyeng-1986-0104

P. Wapperom, Non-isothermal flows of viscoelastic fluids, 1996.

. Wapperon, P. Hulsen-]-wapperon, and M. Hulsen, Thermodynamics of viscoelastic fluids: The temperature equation, Journal of Rheology, vol.42, issue.5, pp.999-1019, 1998.
DOI : 10.1122/1.550922

R. Westover, Instrumentation News: Measuring polymer melt viscosities at high pressures: The hydrostatic pressure rheometer, Advances in Polymer Technology, vol.11, issue.2, pp.147-151, 1992.
DOI : 10.1002/adv.1991.060110208

L. Williams, G. Williams, and H. Lord, Mold-filling studies for the injection molding of thermoplastic materials. Part I: The flow of plastic materials in hot- and cold-walled circular channels, Polymer Engineering and Science, vol.32, issue.8, pp.553-268, 1975.
DOI : 10.1002/pen.760150802

W. Yabe, T. Yabe, and P. Wang, Unified Numerical Procedure for Compressibleand Incompressible Fluid, Journal of the Physical Society of Japan, vol.60, issue.7, pp.2105-2108, 1991.
DOI : 10.1143/JPSJ.60.2105

W. Zienkiewicz, O. Zienkiewicz, and J. Wu, A general explicit or semi-explicit algorithm for compressible and incompressible flows, International Journal for Numerical Methods in Engineering, vol.22, issue.3, pp.475-479, 1992.
DOI : 10.1002/nme.1620350303