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Parametric nonlinear PDEs with multiple solutions: A PGD approach
1. | P' Institute, UPR CNRS - ISAE-ENSMA, 1 avenue Clément Ader, BP 40109, F-86961 Futuroscope Chasseneuil Cedex, France |
2. | GeM Institute, UMR CNRS - Ecole Centrale de Nantes, 1 rue de la Noë, BP 30179, F-44321 Nantes cedex 3, France, France |
References:
[1] |
A. Ammar, M. Normandin, F. Daim, D. Gonzalez, E. Cueto and F. Chinesta, Non incremental strategies based on separated representations: Applications in computational rheology, Communications in Mathematical Sciences, 8 (2010), 671-695.
doi: 10.4310/CMS.2010.v8.n3.a4. |
[2] |
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, 199 (2010), 1872-1880.
doi: 10.1016/j.cma.2010.02.012. |
[3] |
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, 268 (2014), 178-193.
doi: 10.1016/j.cma.2013.09.003. |
[4] |
S. Baguet and B. Cochelin, On the behaviour of the ANM continuation in presence of bifurcations, Communications in Numerical Methods in Engineering, 19 (2003), 459-471.
doi: 10.1002/cnm.605. |
[5] |
E. H. Boutyour, H. Zahrouni, M. Potier-Ferry and M. Boudi, Bifurcation points and bifurcated branches by an asymptotic numerical method and Pade approximants, International Journal for Numerical Methods in Engineering, 60 (2004), 1987-2012.
doi: 10.1002/nme.1033. |
[6] |
F. Chinesta, A. Leygue, F. Bordeu, J. V. Aguado, E. Cueto, D. Gonzalez, I. Alfaro, A. Ammar and A. Huerta, PGD-based computational vademecum for efficient design, optimization and control, Archives of Computational Methods in Engineering, 20 (2013), 31-59.
doi: 10.1007/s11831-013-9080-x. |
[7] |
F. Chinesta, R. Keunings and A. Leygue, The Proper Generalized Decomposition for advanced numerical simulations. A primer, Springerbriefs, Springer, 2014.
doi: 10.1007/978-3-319-02865-1. |
[8] |
B. Cochelin, N. Damil and M. Potier-Ferry, Asymptotic Numerical Methods and Pade approximants for nonlinear elastic structures, International Journal for Numerical Methods in Engineering, 37 (1994), 1187-1213.
doi: 10.1002/nme.1620370706. |
[9] |
C. Ghnatios, F. Chinesta, E. Cueto, A. Leygue, A. Poitou, P. Breitkopf and P. Villon, Methodological approach to efficient modelling and optimization of thermal processes taking place in die: application to pultrusion, Composites Part A, 42 (2011), 1169-1178.
doi: 10.1016/j.compositesa.2011.05.001. |
[10] |
D. Gonzalez, A. Ammar, F. Chinesta and E. Cueto, Recent advances on the use of separated representations, International Journal for Numerical Methods in Engineering, 81 (2010), 637-659.
doi: 10.1002/nme.2710. |
[11] |
H. Herrero, Y. Maday and F. Pla, RB (Reduced Basis) for RB (Rayleigh-Benard), Computer Methods in Applied Mechanics and Engineering, 261/262 (2013), 132-141.
doi: 10.1016/j.cma.2013.02.018. |
[12] |
P. Ladeveze, Nonlinear Computational Structural Mechanics, Springer, New-York, 1999.
doi: 10.1007/978-1-4612-1432-8. |
[13] |
A. Leygue, F. Chinesta, M. Beringhier, T. L. Nguyen, J. C. Grandidier, F. Pasavento and B. Schrefler, Towards a framework for non-linear thermal models in shell domains, International Journal of Numerical Methods for Heat and Fluid Flow, 23 (2013), 53-73.
doi: 10.1108/09615531311289105. |
[14] |
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, 81 (2010), 791-810.
doi: 10.1016/j.matcom.2010.07.015. |
[15] |
J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer-Verlag, 1994.
doi: 10.1007/978-1-4612-0873-0. |
[16] |
F. Terragni and J. M. Vega, On the use of POD-based ROMs to analyze bifurcations in some dissipative systems, Physica D, 241 (2012), 1393-1405.
doi: 10.1016/j.physd.2012.04.009. |
[17] |
P. Vanucci, B. Cochelin, N. Damil and M. Potier-Ferry, An asymptotic-numerical method to compute bifurcating branches, International Journal for Numerical Methods in Engineering, 41 (1998), 1365-1389.
doi: 10.1002/(SICI)1097-0207(19980430)41:8<1365::AID-NME332>3.0.CO;2-Y. |
[18] |
M. Vitse, D. Neron and P. A. Boucard, Virtual charts of solutions for parametrized nonlinear equations, Computational Mechanics, 54 (2014), 1529-1539.
doi: 10.1007/s00466-014-1073-6. |
show all references
References:
[1] |
A. Ammar, M. Normandin, F. Daim, D. Gonzalez, E. Cueto and F. Chinesta, Non incremental strategies based on separated representations: Applications in computational rheology, Communications in Mathematical Sciences, 8 (2010), 671-695.
doi: 10.4310/CMS.2010.v8.n3.a4. |
[2] |
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, 199 (2010), 1872-1880.
doi: 10.1016/j.cma.2010.02.012. |
[3] |
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, 268 (2014), 178-193.
doi: 10.1016/j.cma.2013.09.003. |
[4] |
S. Baguet and B. Cochelin, On the behaviour of the ANM continuation in presence of bifurcations, Communications in Numerical Methods in Engineering, 19 (2003), 459-471.
doi: 10.1002/cnm.605. |
[5] |
E. H. Boutyour, H. Zahrouni, M. Potier-Ferry and M. Boudi, Bifurcation points and bifurcated branches by an asymptotic numerical method and Pade approximants, International Journal for Numerical Methods in Engineering, 60 (2004), 1987-2012.
doi: 10.1002/nme.1033. |
[6] |
F. Chinesta, A. Leygue, F. Bordeu, J. V. Aguado, E. Cueto, D. Gonzalez, I. Alfaro, A. Ammar and A. Huerta, PGD-based computational vademecum for efficient design, optimization and control, Archives of Computational Methods in Engineering, 20 (2013), 31-59.
doi: 10.1007/s11831-013-9080-x. |
[7] |
F. Chinesta, R. Keunings and A. Leygue, The Proper Generalized Decomposition for advanced numerical simulations. A primer, Springerbriefs, Springer, 2014.
doi: 10.1007/978-3-319-02865-1. |
[8] |
B. Cochelin, N. Damil and M. Potier-Ferry, Asymptotic Numerical Methods and Pade approximants for nonlinear elastic structures, International Journal for Numerical Methods in Engineering, 37 (1994), 1187-1213.
doi: 10.1002/nme.1620370706. |
[9] |
C. Ghnatios, F. Chinesta, E. Cueto, A. Leygue, A. Poitou, P. Breitkopf and P. Villon, Methodological approach to efficient modelling and optimization of thermal processes taking place in die: application to pultrusion, Composites Part A, 42 (2011), 1169-1178.
doi: 10.1016/j.compositesa.2011.05.001. |
[10] |
D. Gonzalez, A. Ammar, F. Chinesta and E. Cueto, Recent advances on the use of separated representations, International Journal for Numerical Methods in Engineering, 81 (2010), 637-659.
doi: 10.1002/nme.2710. |
[11] |
H. Herrero, Y. Maday and F. Pla, RB (Reduced Basis) for RB (Rayleigh-Benard), Computer Methods in Applied Mechanics and Engineering, 261/262 (2013), 132-141.
doi: 10.1016/j.cma.2013.02.018. |
[12] |
P. Ladeveze, Nonlinear Computational Structural Mechanics, Springer, New-York, 1999.
doi: 10.1007/978-1-4612-1432-8. |
[13] |
A. Leygue, F. Chinesta, M. Beringhier, T. L. Nguyen, J. C. Grandidier, F. Pasavento and B. Schrefler, Towards a framework for non-linear thermal models in shell domains, International Journal of Numerical Methods for Heat and Fluid Flow, 23 (2013), 53-73.
doi: 10.1108/09615531311289105. |
[14] |
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, 81 (2010), 791-810.
doi: 10.1016/j.matcom.2010.07.015. |
[15] |
J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer-Verlag, 1994.
doi: 10.1007/978-1-4612-0873-0. |
[16] |
F. Terragni and J. M. Vega, On the use of POD-based ROMs to analyze bifurcations in some dissipative systems, Physica D, 241 (2012), 1393-1405.
doi: 10.1016/j.physd.2012.04.009. |
[17] |
P. Vanucci, B. Cochelin, N. Damil and M. Potier-Ferry, An asymptotic-numerical method to compute bifurcating branches, International Journal for Numerical Methods in Engineering, 41 (1998), 1365-1389.
doi: 10.1002/(SICI)1097-0207(19980430)41:8<1365::AID-NME332>3.0.CO;2-Y. |
[18] |
M. Vitse, D. Neron and P. A. Boucard, Virtual charts of solutions for parametrized nonlinear equations, Computational Mechanics, 54 (2014), 1529-1539.
doi: 10.1007/s00466-014-1073-6. |
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