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April  2016, 9(2): 383-392. doi: 10.3934/dcdss.2016002

Parametric nonlinear PDEs with multiple solutions: A PGD approach

Marianne Beringhier 1, , Adrien Leygue 2, and  Francisco Chinesta 2,

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

Received  March 2015 Revised  October 2015 Published  March 2016

This paper presents some insights into the determination, using the Proper Generalized Decomposition, of multiple solutions of nonlinear parametric partial differential equations. Although the Proper Generalized Decomposition (PGD) is well suited for computing the solution of, possibly nonlinear, parametric problems that vary smoothly with a physical parameter, no work has been achieved for the case of problems that exhibit multiple solutions for some values of a parameter. For two representative cases, we show how an appropriate parametrization, combined to a nonlinear solution procedure can be devised to describe and compute the multiple solutions of a PDE.
Keywords: non-linear parametric PDE, model reduction., PGD.
Mathematics Subject Classification: Primary: 65P30, 68U20; Secondary: 74S3.
Citation: Marianne Beringhier, Adrien Leygue, Francisco Chinesta. Parametric nonlinear PDEs with multiple solutions: A PGD approach. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 383-392. doi: 10.3934/dcdss.2016002
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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