×

zbMATH — the first resource for mathematics

A fast, certified and “tuning free” two-field reduced basis method for the metamodelling of affinely-parametrised elasticity problems. (English) Zbl 1425.74465
Summary: This paper proposes a new reduced basis algorithm for the metamodelling of parametrised elliptic problems. The developments rely on the Constitutive Relation Error (CRE), and the construction of separate reduced order models for the primal variable (displacement) and flux (stress) fields. A two-field greedy sampling strategy is proposed to construct these two fields simultaneously and in an efficient manner: at each iteration, one of the two fields is enriched by increasing the dimension of its reduced space in such a way that the CRE is minimised. This sampling strategy is then used as a basis to construct goal-oriented reduced order modelling. The resulting algorithm is certified and “tuning-free”: the only requirement from the engineer is the level of accuracy that is desired for each of the outputs of the surrogate. It is also shown to be significantly more efficient in terms of computational expense than competing methodologies.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
74B05 Classical linear elasticity
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ryckelynck, D.; Chinesta, F.; Cueto, E.; Ammar, A., On the a priori model reduction: overview and recent developments, Arch. Comput. Methods Eng., 13, 1, 91-128, (2006) · Zbl 1142.76462
[2] Ladevèze, P.; Passieux, J.; 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
[3] Chinesta, F.; Ammar, A.; Cueto, E., 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) · Zbl 1269.65106
[4] Chevreuil, M.; Nouy, A., Model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics, Internat. J. Numer. Methods Engrg., 89, 2, 241-268, (2012) · Zbl 1242.74028
[5] Amsallem, D.; Farhat, C., An interpolation method for adapting reduced-order models and application to aeroelasticity, AIAA J., 46, 7, 1803-1813, (2008)
[6] Xiao, M.; Breitkopf, P.; Knopf-Lenoir, R.; Sidorkiewicz, M.; Villon, P., Model reduction by CPOD and Kriging, Struct. Multidiscip. Optim., 41, 555-574, (2010) · Zbl 1274.90365
[7] Kunisch, K.; Volkwein, S., Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM J. Numer. Anal., 40, 2, 492-515, (2003) · Zbl 1075.65118
[8] Meyer, M.; Matthies, H., Efficient model reduction in non-linear dynamics using the Karhunen-loeve expansion and dual-weighted-residual methods, Comput. Mech., 31, 1, 179-191, (2003) · Zbl 1038.74559
[9] Radermacher, A.; Reese, S., A comparison of projection-based model reduction concepts in the context of nonlinear biomechanics, Arch. Appl. Mech., 83, 8, 1193-1213, (2013) · Zbl 1349.74262
[10] Karhunen, K., Über lineare methoden in der wahrscheinlichkeitsrechnung, vol. 37, (1947), Universitat Helsinki · Zbl 0030.16502
[11] Sirovich, L., Turbulence and the dynamics of coherent structures. part i: coherent structures, Quart. Appl. Math., 45, 3, 561-571, (1987) · Zbl 0676.76047
[12] Berkooz, G.; Holmes, P.; Lumley, J. L., The proper orthogonal decomposition in the analysis of turbulent flows, Ann. Rev. Fluid Mech., 25, 1, 539-575, (1993)
[13] Amsallem, D.; Hetmaniuk, U., Error estimates for Galerkin reduced-order models of the semi-discrete wave equation, ESAIM Math. Model. Numer. Anal., 48, 01, (2014) · Zbl 1290.65087
[14] Kerfriden, P.; Ródenas, J.-J.; Bordas, S.-A., Certification of projection-based reduced order modelling in computational homogenisation by the constitutive relation error, Internat. J. Numer. Methods Engrg., 97, 6, 395-422, (2014) · Zbl 1352.74265
[15] Ryckelynck, D., Hyper-reduction of mechanical models involving internal variables, Internat. J. Numer. Methods Engrg., 77, 1, 75-89, (2008) · Zbl 1195.74299
[16] Carlberg, K.; Bou-Mosleh, C.; Farhat, C., Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations, Internat. J. Numer. Methods Engrg., 86, 2, 155-181, (2011) · Zbl 1235.74351
[17] Constantine, P.; Wang, Q., Residual minimizing model interpolation for parameterized nonlinear dynamical systems, SIAM J. Sci. Comput., 34, 4, A2118-A2144, (2012) · Zbl 1259.65110
[18] Kerfriden, P.; Passieux, J.-C.; Bordas, S. P.-A., Local/global model order reduction strategy for the simulation of quasi-brittle fracture, Internat. J. Numer. Methods Engrg., 89, 2, 154-179, (2012) · Zbl 1242.74130
[19] Kerfriden, P.; Goury, O.; Rabczuk, T.; Bordas, S.-A., A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics, Comput. Methods Appl. Mech. Engrg., 200, 5, 850-866, (2012) · Zbl 1352.74285
[20] Fritzen, F.; Leuschner, M., Reduced basis hybrid computational homogenization based on a mixed incremental formulation, Comput. Methods Appl. Mech. Engrg., 260, 143-154, (2013) · Zbl 1286.74081
[21] Prud’homme, C.; Rovas, D. V.; Veroy, K.; Machiels, L.; Maday, Y.; Patera, A. T.; Turinici, G., Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods, J. Fluids Eng., 124, 1, 70-80, (2002)
[22] Rozza, G.; Huynh, D.; Patera, A. T., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Archiv. Comput. Methods Eng., 15, 3, 229-275, (2008) · Zbl 1304.65251
[23] Haasdonk, B.; Ohlberger, M., Reduced basis method for finite volume approximations of parametrized linear evolution equations, ESAIM Math. Model. Numer. Anal., 42, 02, 277-302, (2008) · Zbl 1388.76177
[24] Hoang, K. C.; Kerfriden, P.; Khoo, B. C.; Bordas, S. P.A., An efficient goal-oriented sampling strategy using reduced basis method for parametrized elastodynamic problems, Numer. Methods Partial Differential Equations, 31, 2, 575-608, (2015) · Zbl 1325.74143
[25] Amsallem, D.; Hetmaniuk, U., A posteriori error estimators for linear reduced-order models using Krylov-based integrators, Internat. J. Numer. Methods Engrg., 102, 5, 1238-1261, (2014) · Zbl 1352.65206
[26] Paul-Dubois-Taine, A.; Amsallem, D., An adaptive and efficient greedy procedure for the optimal training of parametric reduced-order models, Internat. J. Numer. Methods Engrg., 102, 5, 1262-1292, (2014) · Zbl 1352.65217
[27] Bui-Thanh, T.; Willcox, K.; Ghattas, O.; van Bloemen Waanders, B., Goal-oriented, model-constrained optimization for reduction of large-scale systems, J. Comput. Phys., 224, 2, 880-896, (2007) · Zbl 1123.65081
[28] Quarteroni, A.; Rozza, G.; Manzoni, A., Certified reduced basis approximation for parametrized partial differential equations and applications, J. Math. Ind., 1, 1, 1-49, (2011) · Zbl 1273.65148
[29] Huynh, D. B.P.; Rozza, G.; Sen, S.; Patera, A. T., A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants, C. R. Math., 345, 8, 473-478, (2007) · Zbl 1127.65086
[30] Huynh, D.; Knezevic, D.; Chen, Y.; Hesthaven, J. S.; Patera, A., A natural-norm successive constraint method for inf-sup lower bounds, Comput. Methods Appl. Mech. Engrg., 199, 29, 1963-1975, (2010) · Zbl 1231.76208
[31] Ladevèze, P.; Pelle, J.-P., Mastering calculations in linear and non linear mechanics, (2004), Springer Verlag New York
[32] Willcox, K.; Peraire, J., Balanced model reduction via the proper orthogonal decomposition, AIAA J., 40, 11, (2002)
[33] A. Janon, M. Nodet, C. Prieur, Goal-oriented error estimation for reduced basis method, with application to certified sensitivity analysis, arXiv preprint arXiv:1303.6618, 2013. · Zbl 1344.65105
[34] Ladevèze, P.; Chamoin, L., On the verification of model reduction methods based on the proper generalized decomposition, Comput. Methods Appl. Mech. Engrg., 200, 23, 2032-2047, (2011) · Zbl 1228.76089
[35] Ryckelynck, D.; Gallimard, L.; Jules, S., Estimation of the validity domain of hyper-reduction approximations in generalized standard elastoviscoplasticity, Adv. Model. Simul. Eng. Sci., 2, 6, (2015)
[36] Gunzburger, M. D.; Peterson, J. S.; Shadid, J. N., Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data, Comput. Methods Appl. Mech. Engrg., 196, 4, 1030-1047, (2007) · Zbl 1121.65354
[37] Hoang, K.; Khoo, B.; Liu, G.; Nguyen, N. C.; Patera, A. T., Rapid identification of material properties of the interface tissue in dental implant systems using reduced basis method, Inverse Probl. Sci. Eng., 21, 8, 1310-1334, (2013) · Zbl 1300.92047
[38] Saad, Y., Iterative methods for sparse linear systems, (2003), SIAM · Zbl 1002.65042
[39] Oden, J. T.; Prudhomme, S., Goal-oriented error estimation and adaptivity for the finite element method, Comput. Math. Appl., 41, 5, 735-756, (2001) · Zbl 0987.65110
[40] Becker, R.; Rannacher, R., An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 10, 1-102, (2003)
[41] Ladevèze, P., Upper error bounds on calculated outputs of interest for linear and nonlinear structural problems, C. R. Math., 334, 7, 399-407, (2006) · Zbl 1177.74367
[42] Stein, E.; Rüter, M., Finite element methods for elasticity with error-controlled discretization and model adaptivity, Encyclopedia Comput. Mech., (2007)
[43] Dıez, P.; Parés, N.; Huerta, A., Error estimation and quality control, Encyclopedia Aerosp. Eng., (2010)
[44] González-Estrada, O.; Nadal, E.; Ródenas, J. J.; Kerfriden, P.; Bordas, S.-A.; Fuenmayor, F. J., Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery, Comput. Mech., 53, 5, 957-976, (2014) · Zbl 1398.74332
[45] Nemat-Nasser, S.; Hori, M., Micromechanics: overall properties of heterogeneous materials, vol. 2, (1999), Elsevier Amsterdam · Zbl 0924.73006
[46] Zohdi, T. I.; Wriggers, P., An introduction to computational micromechanics, vol. 20, (2008), Springer · Zbl 1143.74002
[47] M. Tschopp, Mathworks website: Synthetic microstructure generator. http://www.mathworks.co.uk/matlabcentral/fileexchange/25389-synthetic-microstructure-generator, Nov. 2009.
[48] Tschopp, M.; Wilks, G.; Spowart, J., Multi-scale characterization of orthotropic microstructures, Model. Simul. Mater. Sci. Eng., 16, 6, (2008)
[49] Patera, A. T.
[50] MathWorks, http://www.mathworks.com/help/optim/ug/linprog.html.
[51] TOMLAB. software, http://tomopt.com/tomlab/.
[52] NAG. toolbox for Matlab, http://www.nag.com/numeric/MB/start.asp.
[53] Yue, Y.; Meerbergen, K., Accelerating optimization of parametric linear systems by model order reduction, SIAM J. Optim., 23, 2, 1344-1370, (2013) · Zbl 1273.35279
[54] Zahr, M. J.; Amsallem, D.; Farhat, C., Construction of parametrically-robust CFD-based reduced-order models for PDE-constrained optimization, AIAA paper, vol. 2845, 26-29, (2013)
[55] Cui, T.; Marzouk, Y. M.; Willcox, K. E., Data-driven model reduction for the Bayesian solution of inverse problems, Internat. J. Numer. Methods Engrg., (2014)
[56] Rozza, G., Reduced basis approximation and error bounds for potential flows in parametrized geometries, Commun. Comput. Phys., 9, 1-48, (2011) · Zbl 1284.76295
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.