Abstract
Reduced models and especially those based on Proper Generalized Decomposition (PGD) are decision-making tools which are about to revolutionize many domains. Unfortunately, their calculation remains problematic for problems involving many parameters, for which one can invoke the “curse of dimensionality”. The paper starts with the state-of-the-art for nonlinear problems involving stochastic parameters. Then, an answer to the challenge of many parameters is given in solid mechanics with the so-called “parameter-multiscale PGD”, which is based on the Saint-Venant principle.
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References
A. Ammar, B. Mokdad, F. Chinesta, R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids. Part II: Transient simulation using space-time separated representations. J. Non-Newton. Fluid Mech. 144(2–3), 98–121 (2007)
M. Barrault, Y. Maday, N.C. Nguyen, A.T. Patera, An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. 339(9), 667–672 (2004)
M. Billaud-Friess, A. Nouy, O. Zahm, A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems. ESAIM. Math. Mode. Numer. Anal. 48(6), 1777–1806 (2014)
M. Capaldo, P.-A. Guidault, D. Néron, P. Ladevèze, The Reference Point Method, a "hyperreduction" technique: application to PGD-based nonlinear model reduction. Comput. Method. Appl. Mech. Eng. 322, 483–514 (2017)
S. Chaturantabut, D.C. Sorensen, Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010)
F. Chinesta, P. Ladevèze, (eds.) Separated Representations and PGD-Based Model Reduction: Fundamentals and Applications, vol. CISM 554 (Springer, 2014)
E. de Souza Neto, D. Peric, D.R.J. Owen, Computational Methods for Plasticity, vol. 55 (2008)
A. Falco, W. Hackbusch, A. Nouy, Geometric Structures in Tensor Representations (2015) pp. 1–50 (Work document)
W. Hackbusch, Tensor Spaces and Numerical Tensor Calculus (Springer, 2012)
J.A. Hernandez, J. Oliver, A.E. Huespe, M.A. Caicedo, J.C. Cante, High-performance model reduction techniques in computational multiscale homogenization. Comput. Methods Appl. Mech. Eng. 276, 149–189 (2014)
C. Heyberger, P.A. Boucard, D. Néron, A rational strategy for the resolution of parametrized problems in the PGD framework. Comput. Methods Appl. Mech. Eng. 259, 40–49 (2013)
S. Holtz, T. Rohwedder, R. Schneider, On manifolds of tensors of fixed TT-rank. Numer. Math. 120(4), 701–731 (2012)
P. Ladevèze, On Algorithm Family in Structural Mechanics. Comptes rendus des séances de l’Academie des sciences. Série 2, 300(2) (1985)
P. Ladevèze, The large time increment method for the analyse of structures with nonlinear constitutive relation described by internal variables. Comptes rendus des séances de l’Academie des sciences. Série 2, 309(2), 1095–1099 (1989) (in french)
P. Ladevèze, Nonlinear Computational Structural Mechanics: New Approaches and Non-incremental Methods of Calculation (Springer, New York, 1999)
P. Ladevèze, New variational formulations for discontinuous approximations. Technical Report, LMT Cachan, 2011, (in french)
P. Ladevèze, A new method for the ROM computation: the parameter-multiscale PGD, Technical report, LMT Cachan, 2016a, (in french)
P. Ladevèze, On reduced models in nonlinear solid mechanics. Eur. J. Mech. A/Solids 60, 227–237 (2016b)
P. Ladevèze, J.C. Passieux, D. Néron, The LATIN multiscale computational method and the proper generalized decomposition. Comput. Methods Appl. Mech. Eng. 199(21–22), 1287–1296 (2010)
P. Ladevèze, H. Riou, On Trefftz and weak Trefftz discontinuous Galerkin approaches for medium-frequency acoustics. Comput. Methods Appl. Mech. Eng. 278, 729–743 (2014)
E. Monteiro, J. Yvonnet, Q.C. He, Computational homogenization for nonlinear conduction in heterogeneous materials using model reduction. Comput. Mater. Sci. 42(4), 704–712 (2008)
D. Néron, P.A. Boucard, N. Relun, Time-space PGD for the rapid solution of 3D nonlinear parametrized problems in the many-query context. Int. J. Numer. Methods Eng. 103(4), 275–292 (2015)
A. Nouy, A priori model reduction through proper generalized decomposition for solving time-dependent partial differential equations. Comput. Methods Appl. Mech. Eng. 199(23–24), 1603–1626 (2010)
I.V. Oseledets, Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)
C. Paillet, P. Ladevèze, D. Néron, A Parametric-multiscale PGD for problems with a lage number of parameters (2017) (In preparation)
A. Radermacher, S. Reese, POD-based model reduction with empirical interpolation applied to nonlinear elasticity. Int. J. Numer. Methods Eng. 107(6), 477–495 (2016)
N. Relun, D. Néron, P.A. Boucard, A model reduction technique based on the PGD for elastic-viscoplastic computational analysis. Comput. Mech. 51(1), 83–92 (2013)
D. Ryckelynck, Hyper-reduction of mechanical models involving internal variables. Int. J. Numer. Methods Eng. 77(1), 75–89 (2009)
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Ladevèze, P., Paillet, C., Néron, D. (2018). Extended-PGD Model Reduction for Nonlinear Solid Mechanics Problems Involving Many Parameters. In: Oñate, E., Peric, D., de Souza Neto, E., Chiumenti, M. (eds) Advances in Computational Plasticity. Computational Methods in Applied Sciences, vol 46. Springer, Cham. https://doi.org/10.1007/978-3-319-60885-3_10
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