A reduced basis method for parametrized variational inequalities applied to contact mechanics. (English) Zbl 07843242
Summary: We investigate new developments of the reduced-basis method for parametrized optimization problems with nonlinear constraints. We propose a reduced-basis scheme in a saddle-point form combined with the Empirical Interpolation Method to deal with the nonlinear constraint. In this setting, a primal reduced-basis is needed for the primal solution and a dual one is needed for the Lagrange multipliers. We suggest to construct the latter using a cone-projected greedy algorithm that conserves the non-negativity of the dual basis vectors. The reduction strategy is applied to elastic frictionless contact problems including the possibility of using nonmatching meshes. The numerical examples confirm the efficiency of the reduction strategy.
{© 2019 John Wiley & Sons, Ltd.}
{© 2019 John Wiley & Sons, Ltd.}
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 49Jxx | Existence theories in calculus of variations and optimal control |
| 35Kxx | Parabolic equations and parabolic systems |
Keywords:
constrained problems; contact mechanics; nonlinear model reduction; noninterpenetration condition; nonmatching meshes; variational inequalitiesSoftware:
CVXOPTReferences:
| [1] | HesthavenJS, RozzaG, StammB. Certified Reduced Basis Methods for Parametrized Partial Differential Equations. SpringerBriefs in Mathematics. Cham, Switzerland/Bilbao, Spain: Springer/BCAM Basque Center for Applied Mathematics; 2016. · Zbl 1329.65203 |
| [2] | QuarteroniA, ManzoniA, NegriF. Reduced Basis Methods for Partial Differential Equations. La Matematica per il 3+2. New York, NY: Springer International Publishing; 2016. · Zbl 1337.65113 |
| [3] | HaasdonkB, SalomonJ, WohlmuthB. A reduced basis method for parametrized variational inequalities. SIAM J Numer Anal. 2012;50(5):2656‐2676. · Zbl 1261.65069 |
| [4] | BalajewiczM, AmsallemD, FarhatC. Projection‐based model reduction for contact problems. Int J Numer Methods Eng. 2016;106(8):644‐663. · Zbl 1352.74196 |
| [5] | LeeDD, SeungHS. Algorithms for non‐negative matrix factorization. In: Proceedings of the 13th International Conference on Neural Information Processing Systems, NIPS’00, (Cambridge, MA), MIT Press; 2000:535-541. |
| [6] | FauqueJ, RamièreI, RyckelynckD. Hybrid hyper‐reduced modeling for contact mechanics problems. Int J Numer Methods Eng. 2018;115(1):117‐139. · Zbl 07864831 |
| [7] | GlasS, UrbanK. On noncoercive variational inequalities. SIAM J Numer Anal. 2014;52:2250‐2271. · Zbl 1317.35136 |
| [8] | GlasS, UrbanK. Numerical investigations of an error bound for reduced basis approximations of noncoercice variational inequalities. IFAC‐PapersOnLine. 2015;48(12):721‐726. |
| [9] | ZhangZ, BaderE, VeroyK. A duality approach to error estimation for variational inequalities. arXiv; 2014. |
| [10] | BaderE, ZhangZ, VeroyK. An empirical interpolation approach to reduced basis approximations for variational inequalities. Math Comput Model Dyn Syst. 2016;22(4):345‐361. · Zbl 1351.49034 |
| [11] | ZhangZ, BaderE, VeroyK. A slack approach to reduced‐basis approximation and error estimation for variational inequalities. C R Math Acad Sci Paris. 2016;354(3):283‐289. · Zbl 1387.65059 |
| [12] | BurkovskaO. Reduced Basis Methods for Option Pricing and Calibration [PhD thesis]. Technische Universität München; 2016. |
| [13] | BurkovskaO, HaasdonkB, SalomonJ, WohlmuthB. Reduced basis methods for pricing options with the Black-Scholes and Heston models. SIAM J Financial Math. 2014;6:08. |
| [14] | HaasdonkB, SalomonJ, WohlmuthB. A reduced basis method for the simulation of American options. Numerical Mathematics and Advanced Applications. 2013;2011:821‐829. · Zbl 1269.91086 |
| [15] | GiacomaA, DureisseixD, GravouilA, RochetteM. Vers une approche PGD quasi‐optimale pour les problèmes nonlinéaires de contact frottant. In: 12e Colloque national en calcul des structures, Giens, France, CSMA; 2015. |
| [16] | BarraultM, MadayY, NguyenNC, PateraAT. An ’empirical interpolation’ method: application to efficient reduced‐basis discretization of partial differential equations. C R Math Acad Sci Paris. 2004;339(9):667‐672. · Zbl 1061.65118 |
| [17] | MadayY, NguyenNC, PateraAT, PauGSH. A general multipurpose interpolation procedure: the magic points. Commun Pure Appl Anal. 2009;8(1):383‐404. · Zbl 1184.65020 |
| [18] | HertzH. Über die Berührung fester elastischer Körper. Journal für die reine und angewandte Mathematik. 1882;1882(92):156‐171. · JFM 14.0807.01 |
| [19] | WriggersP. Computational Contact Mechanics. Berlin, Heidelberg: Springer; 2006. · Zbl 1104.74002 |
| [20] | ErnA, GuermondJ‐L. Theory and Practice of Finite Elements. Applied Mathematical Sciences. Vol 159. New York, NY: Springer‐Verlag; 2004. · Zbl 1059.65103 |
| [21] | FučíkS, KratochvílA, NečasJ. Kačanov‐Galerkin method and its application. Acta Univ Carolin Math Phys. 1974;15(1):31‐33. · Zbl 0341.65080 |
| [22] | BenaceurA. Model Reduction for Nonlinear Thermics and Mechanics [PhD thesis]. University Paris‐Est Marne la Vallée; 2018. |
| [23] | HaasdonkB. Convergence rates of the POD‐greedy method. ESAIM Math Model Numer Anal. 2013;47(3):859‐873. · Zbl 1277.65074 |
| [24] | HinzeM, VolkweinS. Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: error estimates and suboptimal control. In: BennerP (ed.), SorensenDC (ed.) & MehrmannV (ed.), eds. Dimension Reduction of Large‐scale Systems. Lecture Notes in Computational Science and Engineering. Vol 45. Berlin, Germany: Springer; 2005:261���306. · Zbl 1079.65533 |
| [25] | KunischK, VolkweinS. Galerkin proper orthogonal decomposition methods for parabolic problems. Numer Math. 2001;90(1):117‐148. · Zbl 1005.65112 |
| [26] | AndersenMS, DahlJ, VandenbergheL. CVXOPT: A Python Package for Convex Optimization; 2008. http://www.abel.ee.ucla.edu/cvxopt |
| [27] | HechtF. New developments in freefem++. J Numer Math. 2012;20(3‐4):1‐14. |
| [28] | DrouetG, HildP. An accurate local average contact method for nonmatching meshes. Numer Math. 2017;136(2):467‐502. · Zbl 1371.35082 |
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