A non-intrusive space-time interpolation from compact Stiefel manifolds of parametrized rigid-viscoplastic FEM problems. (English) Zbl 1478.74080

Summary: This work aims to interpolate parametrized reduced order model (ROM) basis constructed via the proper orthogonal decomposition (POD) to derive a robust ROM of the system’s dynamics for an unseen target parameter value. A novel non-intrusive space-time (ST) POD basis interpolation scheme is proposed, for which we define ROM spatial and temporal basis curves on compact Stiefel manifolds. An interpolation is finally defined on a mixed part encoded in a square matrix directly deduced using the spacial part, the singular values and the temporal part, to obtain an interpolated snapshot matrix, keeping track of accurate space and temporal eigenvectors. Moreover, in order to establish a well-defined curve on the compact Stiefel manifold, we introduce a new procedure, the so-called oriented SVD. Such an oriented SVD produces unique right and left eigenvectors for generic matrices, for which all singular values are distinct. It is important to notice that the ST POD basis interpolation does not require the construction and the subsequent solution of a reduced-order FEM model as classically is done. Hence it is avoiding the bottleneck of standard POD interpolation which is associated with the evaluation of the nonlinear terms of the Galerkin projection on the governing equations. As a proof of concept, the proposed method is demonstrated with the adaptation of rigid-thermoviscoplastic finite element ROMs applied to a typical nonlinear open forging metal forming process. Strong correlations of the ST POD models with respect to their associated high-fidelity FEM counterpart simulations are reported, highlighting its potential use for near real-time parametric simulations using off-line computed ROM POD databases.


74S05 Finite element methods applied to problems in solid mechanics
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
53Z05 Applications of differential geometry to physics


Full Text: DOI arXiv


[1] Chenot, J-L, Recent contributions to the finite element modelling of metal forming processes, J Mater Process Technol, 34, 1-4, 9-18 (1992)
[2] Gronostajski, Z.; Pater, Z.; Madej, L.; Gontarz, A.; Lisiecki, L.; Łukaszek-Sołek, A.; Łuksza, J.; Mróz, S.; Muskalski, Z.; Muzykiewicz, W.; Pietrzyk, M.; Śliwa, RE; Tomczak, J.; Wiewiórowska, S.; Winiarski, G.; Zasadziński, J.; Ziółkiewicz, S., Recent development trends in metal forming, Arch Civil Mech Eng, 19, 3, 898-941 (2019)
[3] Francisco, C.; Pierre, L.; Elías, C., A short review on model order reduction based on proper generalized decomposition, Arch Comput Methods Eng, 18, 4, 395-404 (2011)
[4] Allery, C.; Hamdouni, A.; Ryckelynck, D.; Verdon, N., A priori reduction method for solving the two-dimensional Burgers’ equations, Appl Math Comput, 217, 15, 6671-6679 (2011) · Zbl 1211.65130
[5] Holmes, P.; Lumley, JL; Berkooz, G.; Rowley, CW, Turbulence, coherent structures, dynamical systems and symmetry (2012), Cambridge: Cambridge University Press, Cambridge · Zbl 1251.76001
[6] Henri T, Yvon JP (2005) Convergence estimates of POD-Galerkin methods for parabolic problems. In: IFIP international federation for information processing. Kluwer Academic Publishers, pp 295-306 · Zbl 1088.65576
[7] Nadine, A., On the hidden beauty of the proper orthogonal decomposition, Theoret Comput Fluid Dyn, 2, 5-6, 339-352 (1991) · Zbl 0732.76044
[8] Kari K (1946) Zur spektraltheorie stochastischer prozesse. Ann Acad Sci Fennicae AI 34 · Zbl 0030.20103
[9] Loève, M., Elementary probability theory, 1-52 (1977), New York: Probability theory I. Springer, New York · Zbl 0359.60001
[10] Golub GH, Loan CFV (1996) Matrix computations, vol 1, 3rd edn. JHU Press · Zbl 0865.65009
[11] Jolliffe IT (2002) Springer series in statistics. Principal Comp Anal 29
[12] Hervé, A.; Williams Lynne, J., Principal component analysis, Wiley Interdiscip Rev Comput Stat, 2, 4, 433-459 (2010)
[13] Edward, JJ, Principal components and factor analysis: part I principal components, J Qual Technol, 12, 4, 201-213 (1980)
[14] Edward, JJ, Principal components and factor analysis: part II-additional topics related to principal components, J Qual Technol, 13, 1, 46-58 (1981)
[15] Patricia, A.; Siep, W.; Karen, W.; Ton, B., Missing point estimation in models described by proper orthogonal decomposition, IEEE Trans Autom Control, 53, 10, 2237-2251 (2008) · Zbl 1367.93110
[16] Annika, R.; Stefanie, R., POD-based model reduction with empirical interpolation applied to nonlinear elasticity, Int J Numer Methods Eng, 107, 6, 477-495 (2015) · Zbl 1352.74048
[17] Saifon, C.; Sorensen Danny, C., Nonlinear model reduction via discrete empirical interpolation, SIAM J Sci Comput, 32, 5, 2737-2764 (2010) · Zbl 1217.65169
[18] Everson, R.; Sirovich, L., Karhunen-loève procedure for gappy data, J Opt Soc Am A, 12, 8, 1657 (1995)
[19] Kevin, C.; Charbel, F.; Julien, C.; David, A., The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows, J Opt Soc Am A, 242, 623-647 (2013) · Zbl 1299.76180
[20] David, A.; Julien, C.; Kevin, C.; Charbel, F., A method for interpolating on manifolds structural dynamics reduced-order models, Int J Numer Methods Eng, 80, 9, 1241-1258 (2009) · Zbl 1176.74077
[21] Meza RM (2018) Interpolation sur les variétés grassmanniennes et applications à la réduction de modèles en mécanique. PhD thesis, La Rochelle
[22] Silvère, B.; Rodolphe, S., Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank, SIAM J Matrix Anal Appl, 31, 3, 1055-1070 (2010) · Zbl 1220.47025
[23] Lee JM (2013) Smooth manifolds. In: Introduction to Smooth manifolds. Springer, pp 1-31
[24] Sylvestre, G.; Dominique, H.; Jacques, L., Riemannian geometry (1990), New York: Springer, New York · Zbl 0716.53001
[25] Rolando, M.; Aziz, H.; Abdallah, EH; Cyrille, A., POD basis interpolation via inverse distance weighting on Grassmann manifolds, Discrete Contin Dyn Syst, 12, 6, 1743-1759 (2019) · Zbl 1464.65056
[26] Muhlbach, G., The general Neville-Aitken-algorithm and some applications, Numer Math, 31, 1, 97-110 (1978) · Zbl 0427.65003
[27] Lu, Y.; Blal, N.; Gravouil, A., Space-time POD based computational vademecums for parametric studies: application to thermo-mechanical problems, Adv Model Simul Eng Sci, 5, 1, 1-27 (2018)
[28] Oulghelou M, Allery C (2021) Non intrusive method for parametric model order reduction using a bi-calibrated interpolation on the Grassmann manifold. J Comput Phys 426:
[29] Vilas, S.; Elisabeth, L.; Franck, B.; Yannick, H.; Marianna, B., A Galerkin-free model reduction approach for the Navier-Stokes equations, J Comput Phys, 309, 148-163 (2016) · Zbl 1351.76268
[30] Audouze, C.; De Vuyst, F.; Nair, PB, Reduced-order modeling of parameterized PDEs using time-space-parameter principal component analysis, Int J Numer Methods Eng, 80, 8, 1025-1057 (2009) · Zbl 1176.76059
[31] Youngsoo, C.; Kevin, C., Space-time least-squares Petrov-Galerkin projection for nonlinear model reduction, SIAM J Sci Comput, 41, 1, A26-A58 (2019) · Zbl 1405.65140
[32] Youngsoo, C.; Peter, B.; William, A.; Robert, A.; Kevin, H., Space-time reduced order model for large-scale linear dynamical systems with application to Boltzmann transport problems, J Comput Phys, 424 (2021)
[33] Christophe, A.; Florian, DV; Nair Prasanth, B., Nonintrusive reduced-order modeling of parametrized time-dependent partial differential equations, Numer Methods Part Differ Equ, 29, 5, 1587-1628 (2013) · Zbl 1274.65270
[34] Kobayashi, S.; Kobayashi, S.; Oh, SI; Altan, T., Metal forming and the finite-element method (1989), Oxford: Oxford University Press, Oxford
[35] Lee, CH; Kobayashi, S., New solutions to rigid-plastic deformation problems using a matrix method, J Eng Ind, 95, 3, 865-873 (1973)
[36] Kobayashi, S., Rigid-plastic finite element analysis of axisymmetric metal forming processes, Numerical modeling of manufacturing process, 49-65 (1977), New York: ASME, New York
[37] Feng, ZQ; De Saxcé, G., Rigid-plastic implicit integration scheme for analysis of metal forming, Eur J Mech A Solids, 15, 1, 51-66 (1996) · Zbl 0842.73030
[38] Friderikos O (2011) Two-dimensional rigid-plastic fem simulation of metal forming processes in matlab. In: Proceedings of the 4th international conference on manufacturing and materials engineering (ICMMEN), 3-5 October, Thessaloniki, Greece
[39] Alan, E.; Arias Tomás, A.; Smith Steven, T., The geometry of algorithms with orthogonality constraints, SIAM J Matrix Anal Appl, 20, 2, 303-353 (1998) · Zbl 0928.65050
[40] Absil, P-A; Mahony, R.; Sepulchre, R., Riemannian geometry of Grassmann manifolds with a view on algorithmic computation, Acta Applicandae Mathematicae, 80, 2, 199-220 (2004) · Zbl 1052.65048
[41] Kozlov, SE, Geometry of the real Grassmannian manifolds. Parts I, II. Zapiski Nauchnykh Seminarov POMI, 246, 84-107 (1997)
[42] Yung-Chow, W., Differential geometry of Grassmann manifolds, Proc Nat Acad Sci USA, 57, 3, 589 (1967) · Zbl 0154.21404
[43] Berceanu, S., On the geometry of complex Grassmann manifold, its noncompact dual and coherent states, Bulletin Belgian Math Soc Simon Stevin, 4, 2, 205-243 (1997) · Zbl 0947.53021
[44] de Boor C, Ron A (1992) Computational aspects of polynomial interpolation in several variables. Math Comput 58(198):705-705 · Zbl 0767.41003
[45] Rodney, H., The mathematical theory of plasticity (1998), Oxford: Oxford University Press, Oxford
[46] Markov, AA, On variational principles in the theory of plasticity (1948), Providence: Brown University, Providence
[47] Hill R (1948) A variational principle of maximum plastic work in classical plasticity. Quart J Mech Appl Math 1(1):18-28 · Zbl 0035.40902
[48] Oh, SI, Finite element analysis of metal forming processes with arbitrarily shaped dies, Int J Mech Sci, 24, 8, 479-493 (1982) · Zbl 0488.73038
[49] Zienkiewicz, OC; Godbole, PN, Flow of plastic and visco-plastic solids with special reference to extrusion and forming processes, Int J Numer Methods Eng, 8, 1, 1-16 (1974) · Zbl 0271.73038
[50] Ralston, A.; Rabinowitz, P., A first course in numerical analysis (2001), New York: Courier Corporation, New York · Zbl 0976.65001
[51] Dahlquist G, Björck Å (2008) Numerical methods in scientific computing, vol I. Society for Industrial and Applied Mathematics · Zbl 1153.65001
[52] Felippa, CA; Park, KC, Staggered transient analysis procedures for coupled mechanical systems: formulation, Comput Methods Appl Mech Eng, 24, 1, 61-111 (1980) · Zbl 0453.73091
[53] Rebelo, N.; Kobayashi, S., A coupled analysis of viscoplastic deformation and heat transfer-I, Int J Mech Sci, 22, 11, 699-705 (1980) · Zbl 0455.73004
[54] Rebelo, N.; Kobayashi, S., A coupled analysis of viscoplastic deformation and heat transfer-II, Int J Mech Sci, 22, 11, 707-718 (1980) · Zbl 0455.73005
[55] Chen CC (1978) Rigid-plastic finite-element analysis of ring compression. Appl Numer Methods Form Process
[56] van Rooyen, GT; Backofen, WA, A study of interface friction in plastic compression, Int J Mech Sci, 1, 1, 1-27 (1960)
[57] Ryckelynck, D., Hyper-reduction of mechanical models involving internal variables, Int J Numer Methods Eng, 77, 1, 75-89 (2009) · Zbl 1195.74299
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