Bridging proper orthogonal decomposition methods and augmented Newton-Krylov algorithms: an adaptive model order reduction for highly nonlinear mechanical problems. (English) Zbl 1225.74092

Summary: This article describes a bridge between POD-based model order reduction techniques and the classical Newton/Krylov solvers. This bridge is used to derive an efficient algorithm to correct, “on-the-fly”, the reduced order modelling of highly nonlinear problems undergoing strong topological changes. Damage initiation problems tackled via a corrected hyperreduction method are used as an example. It is shown that the relevancy of reduced order model can be significantly improved with reasonable additional costs when using this algorithm, even when strong topological changes are involved.


74S05 Finite element methods applied to problems in solid mechanics
74S20 Finite difference methods applied to problems in solid mechanics
74R20 Anelastic fracture and damage
74K99 Thin bodies, structures
Full Text: DOI arXiv


[1] Lemaître, J.; Chaboche, J.-L., Mechanics of Solid Materials (1990), Cambridge University Press
[2] Lubineau, G.; Ladevèze, P.; Marsal, D., Towards a bridge between the micro- and mesomechanics of delamination for laminated composites, Compos. Sci. Technol., 66, 6, 698-712 (2007)
[3] Jefferson, A. D.; Bennett, T., A model for cementitious composite materials based on micro-mechanical solutions and damage-contact theory, Comput. Struct. (2009)
[4] Franceschini, G.; Bigoni, D.; Regitnig, P.; Holzapfel, G. A., Brain tissue deforms similarly to filled elastomers and follows consolidation theory, J. Mech. Phys. Solids, 54, 12, 2592-2620 (2006) · Zbl 1162.74303
[5] Moës, N.; Dolbow, J.; Belytschko, T., A finite element method for crack growth without remeshing, Int. J. Engrg. Sci., 46, 131-150 (1999) · Zbl 0955.74066
[6] Bordas, S.; Moran, B., Enriched finite elements and level sets for damage tolerance assessment of complex structures, Engrg. Fract. Mech., 73, 9, 1176-1201 (2006)
[7] Fish, J.; Shek, K.; Pandheeradi, M.; Shephard, M. S., Computational plasticity for composite structures based on mathematical homogenization: theory and practice, Comput. Methods Appl. Mech. Engrg., 148, 53-73 (1997) · Zbl 0924.73145
[8] Feyel, F.; Chaboche, J.-L., \(FE^2\) multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials, Comput. Methods Appl. Mech. Engrg., 183, 309-330 (2000) · Zbl 0993.74062
[9] Gosselet, P.; Rey, C., Non-overlapping domain decomposition methods in structural mechanics, Arch. Comput. Methods Engrg., 13, 515-572 (2006) · Zbl 1171.74041
[10] Ladevèze, P.; Passieux, J. C.; Néron, D., The latin multiscale computational method and the proper generalized decomposition, Comput. Methods Appl. Mech. Engrg., 199, 21, 1287-1296 (2009) · Zbl 1227.74111
[11] Karhunen, K., Über lineare methoden in der wahrscheinlichkeitsrechnung, Ann. Academiæ Scientiarum Fennicæ Series A1, Math. Phys. 37, 37, 1-79 (1947) · Zbl 0030.16502
[12] Loeve, M., Probability Theory (1963), Van Nostrand: Van Nostrand Princeton · Zbl 0108.14202
[13] Ladevèze, P.; Nouy, A., On a multiscale computational strategy with time and space homogenization for structural mechanics, Comput. Methods Appl. Mech. Engrg., 192, 3061-3087 (2003) · Zbl 1054.74701
[14] Yvonnet, J.; He, Q. C., The reduced model multiscale method (r3m) for the non-linear homogenization of hyperelastic media at finite strains, J. Comput. Phys., 223, 1, 341-368 (2007) · Zbl 1163.74048
[15] Pebrel, J.; Rey, C.; Gosselet, P., A nonlinear dual domain decomposition method: application to structural problems with damage, Int. J. Multiscale Comput. Engrg., 6, 3 (2008)
[16] Kerfriden, P.; Allix, O.; Gosselet, P., A three-scale domain decomposition method for the 3D analysis of debonding in laminates, Comput. Mech., 44, 3, 343-362 (2009) · Zbl 1166.74039
[17] Chinesta, F.; Ammar, A.; Cueto, E., Proper generalized decomposition of multiscale models, Int. J. Numer. Methods Engrg. (2009) · Zbl 1197.76093
[18] Ryckelynck, D.; Benziane, D. M., Multi-level a priori hyper-reduction of mechanical models involving internal variables, Comput. Methods Appl. Mech. Engrg., 199, 17-20, 1134-1142 (2010) · Zbl 1227.74093
[19] Liang, Y. C.; Lee, H. P.; Lim, S. P.; Lin, W. Z.; Lee, K. H.; Wu, C. G., Proper orthogonal decomposition and its applications - Part I: Theory, J. Sound Vib., 252, 3, 527-544 (2002) · Zbl 1237.65040
[20] Sirovich, L., Turbulence and the dynamics of coherent structures. Part I: Coherent structures, Q. Appl. Math., 45, 561-571 (1987) · Zbl 0676.76047
[21] Niroomandi, S.; Alfaro, I.; Cueto, E.; Chinesta, F., Real-time deformable models of non-linear tissues by model reduction techniques, Comput. Methods Programs Biomed., 91, 3, 223-231 (2008)
[22] Ryckelynck, D., A priori hyperreduction method: an adaptive approach, J. Comput. Phys., 202, 1, 346-366 (2005) · Zbl 1288.65178
[23] Ryckelynck, D., Hyper-reduction of mechanical models involving internal variables, Int. J. Numer. Methods Engrg., 77, 1, 75-89 (2008) · Zbl 1195.74299
[24] Markovinovic, R.; Jansen, J. D., Accelerating iterative solution methods using reduced-order models as solution predictors, Int. J. Numer. Methods Engrg., 68, 525-554 (2006) · Zbl 1178.76285
[25] Risler, F.; Rey, C., Iterative accelerating algorithms with Krylov subspaces for the solution to large-scale nonlinear problems, Numer. Algorithms, 23, 1-30 (2000) · Zbl 0951.65047
[26] Allix, O.; Kerfriden, P.; Gosselet, P., On the control of the load increments for a proper description of multiple delamination in a domain decomposition framework, Int. J. Numer. Methods Engrg. (2010) · Zbl 1202.74197
[27] Schellenkens, J. C.J.; De Borst, R., On the numerical integration of interface elements, Int. J. Numer. Methods Engrg., 36, 1, 43-66 (1993) · Zbl 0825.73840
[28] Allix, O.; Corigliano, A., Modeling and simulation of crack propagation in mixed-modes interlaminar fracture specimens, Int. J. Fract., 77, 11-140 (1996)
[29] Dostál, Z., Conjugate gradient method with preconditioning by projector, Int. J. Comput. Math., 23, 3, 315-323 (1988) · Zbl 0668.65034
[30] P. Gosselet, Méthodes de décomposition de domaine et méthodes d’accélération pour les problèmes multichamps en mécanique non-linéaire, Ph.D. Thesis, Université Paris 6, 2003.; P. Gosselet, Méthodes de décomposition de domaine et méthodes d’accélération pour les problèmes multichamps en mécanique non-linéaire, Ph.D. Thesis, Université Paris 6, 2003.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.