Abstract
This chapter addresses some advanced topics whose development remains work in progress. The first topic concerns the efficient treatment of non-linear models where standard strategies can fail for high-dimensional problems. The second topic concerns the use of advective stabilization when the involved fields are approximated in a separated form. Finally, we introduce a discrete form of the PGD solver, the one that we consider in computer implementations, that is then extended for considering a separated representation constructor based on residual minimization. Residual minimization is particularly suitable for addressing non-symmetric differential operators, for which the standard procedure described in the previous chapters can be inefficient (slow convergence and non-optimal representations).
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References
A. Ammar, M. Normandin, F. Daim, D. Gonzalez, E. Cueto, F. Chinesta, Non-incremental strategies based on separated representations: applications in computational rheology. Commun. Math. Sci. 8/3, 671–695 (2010)
E. Pruliere, F. Chinesta, A. Ammar, On the deterministic solution of multidimensional parametric models by using the Proper Generalized Decomposition. Math. Comput. Simul. 81, 791–810 (2010)
A. Leygue, F. Chinesta, M. Beringhier, T.L. Nguyen, J.C. Grandidier, F. Pasavento, B. Schrefler, Towards a framework for non-linear thermal models in shell domains. Int. J. Numer. Meth. Heat Fluid Flow 23/1, 55–73 (2013)
B. Cochelin, N. Damil, M. Potier-Ferry, Asymptotic-numerical methods and Pade approximants for non-linear elastic structures. Int. J. Numer. Meth. Eng. 37, 1187–1213 (1994)
B. Cochelin, N. Damil, M. Potier-Ferry, The asymptotic numerical method: an efficient perturbation technique for nonlinear structural mechanics. Revue Europeenne des Elements Finis 3, 281–297 (1994)
M. Barrault, Y. Maday, N.C. Nguyen, A.T. Patera, An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique 339/9, 667–672 (2004)
S. Chaturantabut, D.C. Sorensen, Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32, 2737–2764 (2010)
F. Chinesta, A. Leygue, F. Bordeu, J.V. Aguado, E. Cueto, D. Gonzalez, I. Alfaro, A. Ammar, A. Huerta, PGD-based computational vademecum for efficient design, optimization and control. Arch. Comput. Methods Eng. 20, 31–59 (2013)
J. Donea, A. Huerta, Finite Element Methods for Flow Problems (J Wiley and Sons, Chichester, 2002)
T.J.R. Hughes, A N. Brooks, in A Multidimensional Upwind Scheme with no Crosswind Difusion, ed. by T.J.R. Hughes. Finite Element Methods for Convection Dominated Flows. AMD, vol 34 (American Society of Mechanical Engineering, New York, 1979)
T.J R. Hughes, G.R. Feijóo, L. Mazzei, J-B. Quincy, The variational multiscale method—a paradigm for computational mechanics. Comput. Methods Appl. Mech. Eng. 166/1-2, 3–24 (1998)
D. Gonzalez, E. Cueto, F. Chinesta, P. Diez, A. Huerta, SUPG-based stabilization of proper generalized decompositions for high-dimensional advection-diffusion equations. Int. J. Numer. Meth. Eng. 94/13, 1216–1232 (2013)
F. Chinesta, A. Ammar, E. Cueto, Recent advances and new challenges in the use of the Proper Generalized Decomposition for solving multidimensional models. Arch. Comput. Methods Eng. 17/4, 327–350 (2010)
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Chinesta, F., Keunings, R., Leygue, A. (2014). Advanced Topics. In: The Proper Generalized Decomposition for Advanced Numerical Simulations. SpringerBriefs in Applied Sciences and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-02865-1_6
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DOI: https://doi.org/10.1007/978-3-319-02865-1_6
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