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

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

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.
Citation: Marianne Beringhier, Adrien Leygue, Francisco Chinesta. Parametric nonlinear PDEs with multiple solutions: A PGD approach. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. Ladeveze, Nonlinear Computational Structural Mechanics, Springer, New-York, 1999. doi: 10.1007/978-1-4612-1432-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer-Verlag, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. Ladeveze, Nonlinear Computational Structural Mechanics, Springer, New-York, 1999. doi: 10.1007/978-1-4612-1432-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer-Verlag, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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