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Projection-based reduced order models for a cut finite element method in parametrized domains. (English) Zbl 1443.65348
Summary: This work presents a reduced order modeling technique built on a high fidelity embedded mesh finite element method. Such methods, and in particular the CutFEM method, are attractive in the generation of projection-based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains and in general to handle topological changes. The combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems, avoiding remeshing as well as the reference domain formulation, often used in the reduced order modeling for boundary fitted finite element formulations. The resulting novel methodology is presented on linear elliptic and Stokes problems, together with several test cases to assess its capability. The role of a proper extension and transport of embedded solutions to a common background is analyzed in detail.
MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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[1] Mittal, R.; Iaccarino, G., Immersed boundary methods, Annu. Rev. Fluid Mech., 37, 1, 239-261 (2005) · Zbl 1117.76049
[2] Moes, N.; Dolbow, J.; Belytschko, T., A finite element method for crack growth withought remeshing, Internat. J. Engrg. Sci., 46, 131-150 (1999) · Zbl 0955.74066
[3] Roy, S.; Heltai, L.; Costanzo, F., Benchmarking the immersed finite element method for fluid-structure interaction problems, Comput. Math. Appl., 69, 10, 1167-1188 (2015)
[4] Stein, D.; Guy, R.; Thomases, B., Immersed Boundary Smooth Extension (IBSE): A high-order method for solving incompressible flows in arbitrary smooth domains, J. Comput. Phys., 335, 25-28, 155-178 (2017) · Zbl 1375.76038
[5] Burman, E.; Claus, S.; Hansbo, P.; Larson, M.; Massing, A., CutFEM: Discretizing geometry and partial differential equation, Numer. Methods Eng., 104, 7, 472-501 (2014) · Zbl 1352.65604
[6] Burman, E.; Hansbo, P., Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem, ESAIM: M2AN, 48, 5-8, 859-874 (2014) · Zbl 1416.65437
[7] Burman, E.; Ern, A.; Fernandez, M., Fractional-Step Methods and finite elements with symmetric stabilization for the transient Oseen problem, ESAIM: M2AN, 51, 487-507 (2017) · Zbl 1398.76097
[8] Burman, E.; Fernandez, M., Continuous interior penalty finite element method a for the time-dependent Navier-Stokes equations: space discretization and convergence, Numer. Math., 107, 39-77 (2007) · Zbl 1117.76032
[9] Burman, E.; Fernandez, M. A.; Hansbo, P., Continuous interior penalty finite element a method for Oseen’s equations, Comput. Methods Appl. Mech. Engrg., 44, 3, 1248-1274 (2006) · Zbl 1344.76049
[10] Schott, B.; Wall, W., A new face-oriented stabilized XFEM approach for 2D and 3D incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 276, 233-265 (2014) · Zbl 1423.76273
[11] Liska, S.; T., C., A fast lattice Green’s function method for solving viscous incompressible flows on unbounded domains, J. Comput. Phys., 316, 360-384 (2016) · Zbl 1349.76501
[12] Mengaldo, G.; Liska, S.; Yu, K.; Colonius, T.; Jardin, T., The immersed boundary lattice green function method for external aerodynamics, (23rd AIAA Computational Fluid Dynamics Conference Denver (2017))
[13] Burman, E., Interior penalty variational multiscale method for the incompressible Navier-Stokes equation: Monitoring artificial dissipation, Comput. Methods Appl. Mech. Engrg., 196, 41-44, 4045-4058 (2007) · Zbl 1173.76332
[14] Burman, E.; Fernandez, M., Stabilized explicit coupling for fluid-structure interaction using Nitsche’s method, C. R. Math., 345, 8, 467-472 (2007) · Zbl 1126.74047
[15] Burman, E.; Fernandez, M., Stabilization of explicit coupling in fluid-structure interaction involving fluid incompressibility, Comput. Methods Appl. Mech. Engrg., 198, 5-8, 766-784 (2009) · Zbl 1229.76045
[16] Burman, E.; Fernandez, M., An unfitted nitsche method for incompressible fluid-structure interaction using overlapping meshes, Comput. Methods Appl. Mech. Engrg., 279, 5-8, 497-514 (2014) · Zbl 1423.74867
[17] Court, S.; Fournie, M., A fictitious domain finite element method for simulations of fluid-structure interactions: The Navier-Stokes equations coupled with a moving solid, J. Fluids Struct., 55, 398-408 (2015)
[18] Gerstenberger, A.; Wall, W., An extended Finite Element Method/Lagrange multiplier based approach for fluid-structure interaction, Comput. Methods Appl. Mech. Engrg., 197, 19-20, 1699-1714 (2008) · Zbl 1194.76117
[19] Kallemov, B.; Bhalla, A.; Griffith, B. E.; Donev, A., Immersed boundary method for rigid bodies, Commun. Appl. Math. Comput. Sci., 11, 1, 79-141 (2016) · Zbl 1382.76191
[20] Taira, K.; Colonius, T., The immersed boundary method: a projection approach, J. Comput. Phys., 225, 2, 2118-2137 (2007) · Zbl 1343.76027
[21] Taira, K.; Colonius, T., A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions, Comput. Methods Appl. Mech. Engrg., 197, 25-28, 2131-2146 (2008) · Zbl 1158.76395
[22] Wang, K., A Computational Framework based on an Embedded Boundary Method for Nnonlinear Multi-phase Fluid-structure Interactions (2011), Technical University of Munich, (Ph.D. thesis)
[23] Wang, K.; Lea, P.; Main, A.; McGarity, O.; Farhat, C., Predictive simulation of underwater implosion: Coupling multi-material compressible fluids with cracking structures, (ASME. International Conference on Offshore Mechanics and Arctic Engineering, Volume 8A: Ocean Engineering (2014), American Society of Mechanical Engineers)
[24] Main, A.; Scovazzi, G., The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems, J. Comput. Phys., 372, 972-995 (2018) · Zbl 1415.76457
[25] Main, A.; Scovazzi, G., The shifted boundary method for embedded domain computations. Part II: Linear advection-diffusion and incompressible Navier-Stokes equations, J. Comput. Phys., 372, 996-1026 (2018) · Zbl 1415.76458
[26] Song, T.; Main, A.; Scovazzi, G.; Ricchiuto, M., The shifted boundary method for hyperbolic systems: Embedded domain computations of linear waves and shallow water flows, Inria Bordeaux Sud-Ouest, RR-9136, 1-56 (2017)
[27] Ito, K.; Ravindran, S., A reduced-order method for simulation and control of fluid flows, J. Comput. Phys., 143, 2, 403-425 (1998) · Zbl 0936.76031
[28] Peterson, J., The reduced basis method for incompressible viscous flow calculations, SIAM J. Sci. Stat. Comput., 10, 4, 777-786 (1989) · Zbl 0672.76034
[29] Rozza, G.; Huynh, D.; Patera, A., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: Application to transport and continuum mechanics, Arch. Comput. Methods Eng., 15, 3, 229-275 (2008) · Zbl 1304.65251
[30] Grepl, M.; Patera, A., A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations, ESAIM: M2AN, 39, 1, 157-181 (2005) · Zbl 1079.65096
[31] Grepl, M.; Maday, Y.; Nguyen, N.; Patera, A., Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations, ESAIM: M2AN, 41, 3, 575-605 (2007) · Zbl 1142.65078
[32] Veroy, K.; Prud’homme, C.; Patera, A., Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds, C. R. Math., 337, 9, 619-624 (2003) · Zbl 1036.65075
[33] Ballarin, F.; Manzoni, A.; Quarteroni, A.; Rozza, G., Supremizer stabilization of POD-Galerkin approximation of parametrized steady incompressible Navier-Stokes equations, Internat. J. Numer. Methods Engrg., 102, 5, 1136-1161 (2015) · Zbl 1352.76039
[34] Caiazzo, A.; Iliescu, T.; John, V.; Schyschlowa, S., A numerical investigation of velocity-pressure reduced order models for incompressible flows, J. Comput. Phys., 259, 598-616 (2014) · Zbl 1349.76050
[35] Gerner, A.; Veroy, K., Certified reduced basis methods for parametrized saddle point problems, SIAM J. Sci. Comput., 34, 5, A2812-A2836 (2012) · Zbl 1255.76024
[36] Rozza, G.; Huynh, D. B.P.; Manzoni, A., Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: Roles of the inf-sup stability constants, Numer. Math., 125, 1, 115-152 (2013) · Zbl 1318.76006
[37] Rozza, G.; Veroy, K., On the stability of the reduced basis method for Stokes equations in parametrized domains, Comput. Methods Appl. Mech. Engrg., 196, 7, 1244-1260 (2007) · Zbl 1173.76352
[38] S. Hijazi, S. Ali, G. Stabile, F. Ballarin, G. Rozza, The Effort of Increasing Reynolds Number in Projection-Based Reduced Order Methods: from Laminar to Turbulent Flows, FEF special volume (2017).
[39] Akhtar, I.; Nayfeh, A.; Ribbens, C., On the stability and extension of reduced-order Galerkin models in incompressible flows, Theor. Comput. Fluid Dyn., 23, 3, 213-237 (2009) · Zbl 1234.76040
[40] Bergmann, M.; Bruneau, C.-H.; Iollo, A., Enablers for robust POD models, J. Comput. Phys., 228, 2, 516-538 (2009) · Zbl 1409.76099
[41] L. Fick, Y. Maday, A. Patera, T. Taddei, A Reduced Basis Technique for Long-Time Unsteady Turbulent Flows, arXiv preprint arXiv:1710.03569 (2017).
[42] Iollo, A.; Lanteri, S.; Désidéri, J.-A., Stability properties of POD-Galerkin approximations for the compressible Navier-Stokes equations, Theor. Comput. Fluid Dyn., 13, 6, 377-396 (2000) · Zbl 0987.76077
[43] Sirisup, S.; Karniadakis, G., Stability and accuracy of periodic flow solutions obtained by a POD-penalty method, Physica D, 202, 3-4, 218-237 (2005) · Zbl 1070.35024
[44] Ballarin, F.; Rozza, G., POD-galerkin monolithic reduced order models for parametrized fluid-structure interaction problems, Internat. J. Numer. Methods Fluids, 82, 12, 1010-1034 (2016)
[45] Benner, P.; Ohlberger, M.; Patera, A.; Rozza, G.; Urban, K., (Model Reduction of Parametrized Systems. Model Reduction of Parametrized Systems, MS&A series, vol. 17 (2017), Springer)
[46] Jäggli, C.; Iapichino, L.; Rozza, G., An improvement on geometrical parameterizations by transfinite maps, C. R. Math., 352, 3, 263-268 (2014) · Zbl 1302.35167
[47] C. Lehrenfeld, S. Rave, Mass Conservative Reduced Order Modeling of a Free Boundary Osmotic Cell Swelling Problem, arXiv preprint arXiv:1805.01812 (2018).
[48] Manzoni, A.; Negri, F., Efficient reduction of PDEs defined on domains with variable shape, (Benner, P.; Ohlberger, M.; Patera, A.; Rozza, G.; Urban, K., Model Reduction of Parametrized Systems, Vol. 17 (2017), MS&A book series, Springer), 183-199 · Zbl 1388.78020
[49] Rozza, G., Reduced basis methods for Stokes equations in domains with non-affine parameter dependence, Comput. Vis. Sci., 12, 1, 23-35 (2009)
[50] Ballarin, F.; Faggiano, E.; Ippolito, S.; Manzoni, A.; Quarteroni, A.; Rozza, G.; Scrofani, R., Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization, J. Comput. Phys., 315, 609-628 (2016) · Zbl 1349.76173
[51] Ballarin, F.; D’Amario, A.; Perotto, S.; Rozza, G., A POD-selective inverse distance weighting method for fast parametrized shape morphing, Internat. J. Numer. Methods Fluids, 1-15 (2018)
[52] Ballarin, F.; Manzoni, A.; Rozza, G.; Salsa, S., Shape optimization by Free-Form Deformation: existence results and numerical solution for Stokes flows, J. Sci. Comput., 60, 3, 537-563 (2014) · Zbl 1303.49017
[53] Tezzele, M.; Ballarin, F.; Rozza, G., Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods, (Boffi, D.; Pavarino, L. F.; Rozza, G.; Scacchi, S.; Vergara, C., Mathematical and Numerical Modeling of the Cardiovascular System and Applications (2018), Springer International Publishing), 185-207
[54] Tezzele, M.; Salmoiraghi, F.; Mola, A.; Rozza, G., Dimension reduction in heterogeneous parametric spaces with application to naval engineering shape design problems, Adv. Model. Simul. Eng. Sci., 5:25, 1, 1-19 (2018)
[55] Balajewicz, M.; Farhat, C., Reduction of nonlinear embedded boundary models for problems with evolving interfaces, J. Comput. Phys., 274, 489-504 (2014) · Zbl 1352.65322
[56] Karatzas, E.; Stabile, G.; Atallah, N.; Scovazzi, G.; Rozza, G., A reduced order approach for the embedded shifted boundary FEM and a heat exchange system on parametrized geometries, (IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22-25, 2018 (2018), Springer International Publishing), 111-125 · Zbl 1442.65376
[57] Karatzas, E.; Stabile, G.; Nouveau, L.; Scovazzi, G.; Rozza, G., A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow, Comput. Methods Appl. Mech. Engrg., 347, 568-587 (2019)
[58] E. Karatzas, G. Stabile, L. Nouveau, G. Scovazzi, G. Rozza, A Reduced-Order Shifted Boundary Method for Parametrized Incompressible Navier-Stokes Equations, submitted for publication, arXiv preprint, arXiv:1907.10549 (2019).
[59] E. Karatzas, G. Rozza, Reduced Order Modeling and a stable embedded boundary parametrized Cahn-Hilliard phase field system based on cut finite elements, in preparation (2019).
[60] Bernard, F.; Iollo, A.; Riffaud, S., Reduced-order model for the BGK equation based on POD and optimal transport, J. Comput. Phys., 373, 545-570 (2018) · Zbl 1416.76138
[61] Cagniart, N.; Maday, Y.; Stamm, B., Model order reduction for problems with large convection effects, (Computational Methods in Applied Sciences book series, Vol. 47 (2019), Springer International Publishing), 131-150 · Zbl 1416.35021
[62] Iollo, A.; Lombardi, D., Advection modes by optimal mass transfer, Phys. Rev. E, 89, 022923 (2014)
[63] Naira, N.; Balajewicz, M., Transported snapshot model order reduction approach for parametric, steady-state fluid flows containing parameter dependent shocks, Internat. J. Numer. Methods Fluids, 1-29 (2018)
[64] Reiss, J.; Schulze, P.; Sesterhenn, J.; Mehrmann, V., The shifted proper orthogonal decomposition: A mode decomposition for multiple transport phenomena, SIAM J. Sci. Comput., 40, 3, A1322-A1344 (2018) · Zbl 1446.65212
[65] G. Welper, \(h\) and \(h p\)-adaptive interpolation by transformed snapshots for parametric and stochastic hyperbolic PDEs, arXiv preprint arXiv:1710.11481 (2017). · Zbl 1370.41050
[66] Branets, L.; Ghai, S.; Lyons, S.; Wu, X., Challenges and technologies in reservoir modeling, Commun. Comput. Phys., 6, 1, 1-23 (2009) · Zbl 1364.76216
[67] Burman, E.; Hansbo, P., Fictitious domain finite element methods using cut elements: II. A stabilized nitsche method, Appl. Numer. Math., 52, 6, 2837-2862 (2011)
[68] Antonietti, P. F.; Cangiani, A.; Collis, J.; Dong, Z.; Georgoulis, E. H.; Giani, S.; Houston, P., Review of discontinuous Galerkin finite element methods for partial differential equations on complicated domains, (Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, Lecture Notes in Computational Science and Engineering, vol. 114 (2016), Springer), 279-308
[69] Georgoulis, E. H.; Süli, E., Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method, IMA J. Numer. Anal., 25, 1, 205-220 (2005) · Zbl 1069.65118
[70] Cangiani, A.; Dong, Z.; Georgoulis, E.; Houston, P., (hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes. hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes, SpringerBriefs in Mathematics (2017), Springer International Publishing) · Zbl 1382.65307
[71] Hesthaven, J.; Rozza, G.; Stamm, B., (Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Certified Reduced Basis Methods for Parametrized Partial Differential Equations, SpringerBriefs in Mathematics (2016), Springer International Publishing) · Zbl 1329.65203
[72] Chinesta, F.; Huerta, A.; Rozza, G.; Willcox, K., Encyclopedia of Computational Mechanics, Second Edition, 1-36 (2017), John Wiley & Sons, Ch. Model Reduction Methods
[73] Kalashnikova, I.; Barone, M. F., On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far-field boundary treatment, Internat. J. Numer. Methods Engrg., 83, 10, 1345-1375 (2010) · Zbl 1202.74123
[74] Quarteroni, A.; Manzoni, A.; Negri, F., Reduced Basis Methods for Partial Differential Equations, Vol. 92 (2016), UNITEXT/La Matematica per il 3+2 book series, Springer International Publishing
[75] Chinesta, F.; Ladeveze, P.; Cueto, E., A short review on model order reduction based on proper generalized decomposition, Arch. Comput. Methods Eng., 18, 4, 395 (2011)
[76] Dumon, A.; Allery, C.; Ammar, A., Proper general decomposition (PGD) for the resolution of Navier-Stokes equations, J. Comput. Phys., 230, 4, 1387-1407 (2011) · Zbl 1391.76099
[77] Kunisch, K.; Volkwein, S., Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM J. Numer. Anal., 40, 2, 492-515 (2002) · Zbl 1075.65118
[78] Barrault, M.; Maday, Y.; Nguyen, N.; Patera, A., An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations, C. R. Math., 339, 9, 667-672 (2004) · Zbl 1061.65118
[79] Stabile, G.; Rozza, G., Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier-Stokes equations, Comput. & Fluids, 173, 273-284 (2018) · Zbl 1410.76264
[80] S. Ali, F. Ballarin, G. Rozza, Stabilized reduced basis methods for parametrized steady Stokes and Navier-Stokes equations, submitted for publication (2018).
[81] Stabile, G.; Ballarin, F.; Zuccarino, G.; Rozza, G., A reduced order variational multiscale approach for turbulent flows, Adv. Comput. Math., 1-20 (2019)
[82] Becker, R.; Burman, E.; Hansbo, P., A nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity, Comput. Methods Appl. Mech. Engrg., 198, 3352-3360 (2009) · Zbl 1230.74169
[83] Ballarin, F.; Rozza, G.; Maday, Y., Reduced-order semi-implicit schemes for fluid-structure interaction problems, (Benner, P.; Ohlberger, M.; Patera, A.; Rozza, G.; Urban, K., Model Reduction of Parametrized Systems (2017), MS&A book series, Springer), 149-167 · Zbl 06861097
[84] Jonsson, T.; Larson, M. G.; Larsson, K., Cut finite element methods for elliptic problems on multipatch parametric surfaces, Comput. Methods Appl. Mech. Engrg., 324, 366-394 (2017)
[85] Benamou, J.; Carlier, G.; Cuturi, M.; Nenna, L.; Peyré, G., Iterative bregman projections for regularized transportation problems, SIAM J. Sci. Comput., 37, 2, A1111-A1138 (2015) · Zbl 1319.49073
[86] Solomon, J.; de Goes, F.; Peyré, G.; Cuturi, M.; Butscher, A.; Nguyen, A.; Du, T.; Guibas, L., Convolutional wasserstein distances: Efficient optimal transportation on geometric domains, ACM Trans. Graph., 34, 4, 66:1-66:11 (2015) · Zbl 1334.68267
[87] M. Nonino, F. Ballarin, G. Rozza, Y. Maday, Reduction of the Kolmogorov n-width for a transport dominated fluid-structure interaction problem, in preparation (2019).
[88] G. Stabile, M. Zancanaro, G. Rozza, Efficient Geometrical parametrization for finite-volume based reduced order methods, arXiv preprint, arXiv:1901.06373, 2019.
[89] ngsxfem - Add-On to NGSolve for unfitted finite element discretizations, https://github.com/ngsxfem/ngsxfem, Accessed: 2018-01-30.
[90] J. Schöberl, A. Arnold, J. Erb, J.M. Melenk, T.P. Wihler, \(C+ +11\) implementation of finite elements in NGSolve, Tech. rep., Institute for Analysis and Scientific Computing, Vienna University of Technology, ASC Report 30/2014 (2014).
[91] RBniCS - reduced order modelling in FEniCS, http://mathlab.sissa.it/rbnics, Accessed: 2018-01-30.
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