Hyper-reduced order models for parametrized unsteady Navier-Stokes equations on domains with variable shape

Niccolò Dal Santo; Andrea Manzoni

doi:10.1007/s10444-019-09722-9

Advances in Computational Mathematics

pp 1–39 | Cite as

Hyper-reduced order models for parametrized unsteady Navier-Stokes equations on domains with variable shape

Authors
Authors and affiliations

Niccolò Dal Santo
Andrea Manzoni

Article

First Online: 27 November 2019

5 Downloads

Part of the following topical collections:

Model reduction of parametrized Systems

Abstract

In this work, we set up a new, general, and computationally efficient way to tackle parametrized fluid flows modeled through unsteady Navier-Stokes equations defined on domains with variable shape, when relying on the reduced basis method. We easily describe a domain by flexible boundary parametrizations, and generate domain (and mesh) deformations by means of a solid extension, obtained by solving a harmonic extension or a linear elasticity problem. The proposed procedure is built over a two-stage reduction: (i) first, we construct a reduced basis approximation for the mesh motion problem, irrespective of the fluid flow problem we focus on; (ii) then, we generate a reduced basis approximation of the unsteady Navier-Stokes problem, relying on finite element snapshots evaluated over a set of reduced deformed configuration, and approximating both velocity and pressure fields simultaneously. To deal with unavoidable nonaffine parametric dependencies arising in both the mesh motion and the state problem, we apply a matrix version of the discrete empirical interpolation method, allowing treating a wide range of geometrical deformations in an efficient and purely algebraic way. The same strategy is used to perform hyper-reduction of nonlinear terms. To assess the numerical performances of the proposed technique, we address the solution of parametrized fluid flows where the parameters describe both the shape of the domain and relevant physical features. Complex flow patterns such as the ones appearing in a patient-specific carotid bifurcation are accurately approximated, as well as derived quantities of potential clinical interest.

Keywords

Reduced basis method Parametrized PDEs Hyper-reduction techniques Navier-Stokes equations Computational fluid dynamics Varying domains

Mathematics Subject Classification (2010)

65M60 65N12 76M10

This is a preview of subscription content, to check access.

Preview

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

We acknowledge the Swiss National Supercomputing Centre (CSCS) for providing us the CPU resources under project ID s796.

Funding information

The research of N. Dal Santo has been supported by the Swiss State Secretariat for Education, Research and Innovation (SERI), project No. C14.0068, in the framework of the COST action number TD1307.

A Appendix

A.1 Proper orthogonal decomposition

In this paper, we employ POD both for the construction of RB spaces (see Section 3.1) and for the efficient approximation of μ-dependent arrays (see Section 3.3). For simplicity, here we focus on the former aspect (see, e.g., [43] and the following section for further insights on the latter).

Let us denote by $\mathbf {S} = [\mathbf {s}^{{\boldsymbol {\mu }}_{1}}| {\dots } |\mathbf {s}^{{\boldsymbol {\mu }}_{n_s}}] \in \mathbb {R}^{N_h \times n_s}$ a matrix collecting as columns n_s FE vectors $\{\vec {s}^{\boldsymbol {\mu }_{i}}\}_{i=1}^{N_{s}} \subset \mathbb {R}^{N_h} $ (called snapshots). For any prescribed dimension N, POD provides the N-dimensional subspace, spanned by the columns of $\mathbf {V} = [\xi _{1}|\dots |\xi _{N}] \in \mathbb {R}^{N_h\times N} $, which best approximates $\{\vec {s}^{\boldsymbol {\mu }_{i}}\}_{i=1}^{N_{s}}$ among all possible N-dimensional subspaces. To this goal, POD computes the singular value decomposition (SVD) $ \mathbf {X}^{\frac {1}{2}} \mathbf {S} = \mathbf {U} {\varSigma }\mathbf {Z}^{T}$ of the matrix S, with respect to a scalar product induced by a symmetric positive definite matrix X, where $\mathbf {U} \in \mathbb {R}^{N_h\times N_h} $ and $\mathbf {Z} \in \mathbb {R}^{n_s\times n_s} $ denote orthogonal matrices and $ {\varSigma } \in \mathbb {R}^{N_h\times n_s} $ is a diagonal matrix containing the singular values $\sigma _{1} \geq \sigma _{2} \geq {\dots } \geq \sigma _{n_{s}} \geq 0 $. The matrix V is obtained by retaining the first N columns of U and represents, by construction, an X-orthonormal basis of the best N-dimensional subspace approximating the snapshot set. In particular, we have that:

$$ \begin{array}{@{}rcl@{}} \sum\limits_{i=1}^{n_{s}} \|\mathbf{s}_i - \mathbf{V}\mathbf{V}^{T}\mathbf{X}\mathbf{s}_i \|_{\mathbf{X}}^{2} = \min\limits_{\mathbf{W} \in \mathcal{V}_{N}} \sum\limits_{i=1}^{n_{s}} \|\mathbf{s}_i - \mathbf{W}\mathbf{W}^{T}\mathbf{X}\mathbf{s}_i \|_{\mathbf{X}}^{2} = \sum\limits_{i=N+1}^{n_{s}} {\sigma_{i}^{2}} \end{array} $$

where $\mathcal {V}_{N} = \{\mathbf {W} \in \mathbb {R}^{N_h \times N}: \mathbf {W}^{T}\mathbf {X}\mathbf {W} = \mathbf {I}_{N} \} $ and I_N is the N-dimensional identity matrix.

Moreover, the discarded singular values provide an estimate of the relative error, since:

$$ \begin{array}{@{}rcl@{}} \frac{1}{\sum\limits_{i=1}^{n_{s}} \|\mathbf{s}_i \|_{\mathbf{X}}^{2}} \sum\limits_{i=1}^{n_{s}} \|\mathbf{s}_i - \mathbf{V}\mathbf{V}^{T}\mathbf{X}\mathbf{s}_i \|_{\mathbf{X}}^{2} = \frac{1}{\sum\limits_{i=1}^{n_{s}} {\sigma_{i}^{2}}} \sum\limits_{i=N+1}^{n_{s}} {\sigma_{i}^{2}}. \end{array} $$

(38)

POD is performed by means of Algorithm 3; for a given tolerance ε_POD, (38) is employed to control the relative error on the approximation of the snapshots and to select N basis functions; alternatively, one could directly provide a dimension N instead of ε_POD.

A.2 Matrix DEIM

Because of its pivotal role in enhancing the efficiency of the proposed ROM workflow, here we sketch some details about the way (discrete) empirical interpolation can be used to approximate a matrix $\mathbb {K}(\tau ) \colon \mathcal {T} \mapsto \mathbb {R}^{N_{h} \times N_{h}}$, where τ denotes a parameter vector and/or time.

A Matrix version of DEIM (MDEIM) has been proposed in several works (see, e.g., [12, 65]) to directly approximate a matrix, parameter-dependent operator in a purely algebraic way. A similar idea, denoted as empirical operator interpolation, has been introduced in [19] to treat nonlinear operators in RB approximation of parametrized evolution equations, considering empirical interpolation of discrete nonlinear differential operators and their directional derivatives. Here, we rely on the framework set in [43], tailored to the case of parametrized PDEs in the presence of complex parametrizations.

Given $\mathbb {K}(\tau ) \colon \mathcal {T} \mapsto \mathbb {R}^{N_{h} \times N_{h}}$, MDEIM provides M ≪ N_h functions $\theta _{q} \colon \mathcal {T} \mapsto \mathbb {R}$ and τ-independent matrices $\mathbb {K}_{q} \in \mathbb {R}^{N_{h} \times N_{h}}$, 1 ≤ q ≤ M, such that:

$$ \mathbb{K}(\tau) \approx \mathbb{K}_{m} (\tau) = {\sum}_{q=1}^{M} \theta_{q}(\tau) \mathbb{K}_{q}. $$

(39)

To avoid misunderstandings, here we adopt a different notation from the one used in Section A.1, although some operations are indeed very similar.

The offline stage of MDEIM consists of two main steps. First, we express $\mathbb {K}(\tau )$ in vector format by stacking its columns, that is, we set $\mathbf {k}(\tau ) = \text {vec}(\mathbb {K}(\tau )) \in \mathbb {R}^{{N_{h}^{2}}}$. Hence, (39) can be reformulated as: find {Φ, 𝜃(τ)} such that:

$$ \mathbf{k}(\tau) \approx \mathbf{k}_{m}(\tau) = {\varPhi} {\theta}(\tau), $$

(40)

where $ {\varPhi } \in \mathbb {R}^{{N_{h}^{2}} \times M}$ is a τ-independent basis and $ {\theta }(\tau ) \in \mathbb {R}^{M}$ the corresponding coefficient vector.

Then, we apply DEIM [13] to a set of snapshots $\textbf {S}\! = \! [ \text {vec}(\mathbb {K}(\tau _{1})), \dots ,\text {vec}(\mathbb {K}(\tau _{n_{s}}))]$ in order to obtain the basis Φ and a set of interpolation indices $\mathcal {I} \subset \{1, \cdots , {N_{h}^{2}}\}$. The former is computed by applying POD (for a given tolerance δ_mdeim) over the columns of S, whereas the latter is iteratively selected by employing the magic points algorithm [36]. Both these steps are reported in Algorithm 4.

During the online phase, given a new $\tau \in \mathcal {T}$, we can compute $\mathbb {K}_{m}(\tau )$ as

$$ \mathbb{K}_{m}(\tau) = \text{vec}^{-1}(\boldsymbol{\varPhi} \boldsymbol{\theta}(\tau)) \quad \text{with } \quad \boldsymbol{\theta}(\tau) \ \text{such that} \ \boldsymbol{\varPhi}_{\mathcal{I}} \boldsymbol{\theta}(\tau) = \mathbb{K}_{\mathcal{I}}(\tau), $$

(41)

where $\boldsymbol {\varPhi }_{\mathcal {I}}$ and $\mathbb {K}_{\mathcal {I}}(\tau )$ denote the matrices formed by the $\mathcal {I}$ rows of Φ and $\mathbb {K}(\tau )$, respectively.

Note that evaluating $\mathbb {K}_{\mathcal {I}}(\tau )$ online can be performed efficiently when $\mathbb {K}(\tau )$ results from a FE discretization of a PDE operator, by employing the same assembly routine used for the FOM on the reduced mesh associated to the selected interpolation indices $\mathcal {I}$ (see, e.g., [8, 43]).

References

1.
Baiges, J., Codina, R., Idelsohn, S.: Explicit reduced-order models for the stabilized finite element approximation of the incompressible Navier-Stokes equations. Int. J. Numer. Meth Fluids 72(12), 1219–1243 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
2.
Baker, T.J.: Mesh movement and metamorphosis. Eng. Comput. 18(3), 188–198 (2002)CrossRefGoogle Scholar
3.
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)MathSciNetzbMATHCrossRefGoogle Scholar
4.
Ballarin, F., Faggiano, E., Manzoni, A., Quarteroni, A., Rozza, G., Ippolito, S., Antona, C., Scrofani, R.: Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts Biomech. Model Mechan. 16(4), 1373–1399 (2017)zbMATHCrossRefGoogle Scholar
5.
Ballarin, F., Manzoni, A., Quarteroni, A., Rozza, G.: Supremizer stabilization of P,OD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations. Int. J. Numer Methods Engng. 102(5), 1136–1161 (2015)zbMATHCrossRefGoogle Scholar
6.
Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
7.
Bergmann, M., Bruneau, C.-H., Iollo, A.: Enablers for robust POD models. J. Comput. Phys. 228(2), 516–538 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
8.
Bonomi, D., Manzoni, A., Quarteroni, A.: A matrix DEIM technique for model reduction of nonlinear parametrized problems in cardiac mechanics. Comput. Methods Appl. Mech. Engrg. 324, 300–326 (2017)MathSciNetCrossRefGoogle Scholar
9.
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)MathSciNetzbMATHCrossRefGoogle Scholar
10.
Campbell, I.C., Ries, J., Dhawan, S.S., Quyyumi, A.A., Taylor, W.R., Oshinski, J.N.: Effect of inlet velocity profiles on patient-specific computational fluid dynamics simulations of the carotid bifurcation. J. Biomech. Engng. 134(5), 051001 (2012)CrossRefGoogle Scholar
11.
Carlberg, K., Farhat, C., Cortial, J., Amsallem, D.: The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows. J. Comput. Phys. 242, 623–647 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
12.
Carlberg, K., Tuminaro, R., Boggs, P.: Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics. SIAM J. Sci. Comput. 37(2), B153–B184 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
13.
Chaturantabut, S.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
14.
Colciago, C.M., Deparis, S., Quarteroni, A.: Comparisons between reduced order models and full 3d models for fluid–structure interaction problems in haemodynamics. J. Comput. Appl. Math. 265, 120–138 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
15.
Dal Santo, N., Deparis, S., Manzoni, A., Quarteroni, A.: An algebraic least squares reduced basis method for the solution of parametrized Stokes equations. Technical Report 21.2017 MATHICSE–EPFL (2017)Google Scholar
16.
De Sturler, E., Liesen, J.: Block-diagonal and constraint preconditioners for nonsymmetric indefinite linear systems. part i: Theory. SIAM J. Sci. Comput. 26(5), 1598–1619 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
17.
Deparis, S.: Reduced basis error bound computation of parameter-dependent Navier–Stokes equations by the natural norm approach. SIAM J. Numer. Anal. 46 (4), 2039–2067 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
18.
Drohmann, M., Haasdonk, B., Ohlberger, M.: Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput. 34(2), A937–A969 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
19.
Drohmann, M., Haasdonk, B., Ohlberger, M.: Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput. 34(2), A937–A969 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
20.
Díez, P., Zlotnik, S., Huerta, A.: Generalized parametric solutions in Stokes flow. Comput. Methods Appl. Mech. Engrg. 326, 223–240 (2017)MathSciNetCrossRefGoogle Scholar
21.
Elman, H., Howle, V.E., Shadid, J., Shuttleworth, R., Tuminaro, R.: Block preconditioners based on approximate commutators. SIAM J. Sci Comput. 27 (5), 1651–1668 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
22.
Elman, H.C., Forstall, V.: Numerical solution of the parameterized steady-state Navier-Stokes equations using empirical interpolation methods. Comput. Methods Appl. Mech. Engrg. 317, 380–399 (2017)MathSciNetCrossRefGoogle Scholar
23.
Elman, H.C., Silvester, D.: Fast nonsymmetric iterations and preconditioning for Navier–Stokes equations. SIAM J. Sci. Comput. 17(1), 33–46 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
24.
Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics (2005)Google Scholar
25.
Forti, D., Dedè, L.: Semi-implicit bdf time discretization of the Navier–Stokes equations with vms-les modeling in a high performance computing framework. Comput. Fluids 117, 168–182 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
26.
Gervasio, P., Saleri, F., Veneziani, A.: Algebraic fractional-step schemes with spectral methods for the incompressible navier–stokes equations. J. Comput. Phys. 214(1), 347–365 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
27.
Ghia, U., Ghia, K.N., Shin, C.: High-re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys. 48(3), 387–411 (1982)zbMATHCrossRefGoogle Scholar
28.
Guerciotti, B., Vergara, C., Azzimonti, L., Forzenigo, L., Buora, A., Biondetti, P., Domanin, M.: Computational study of the fluid-dynamics in carotids before and after endarterectomy. J. Biomech. 49(1), 26–38 (2016)CrossRefGoogle Scholar
29.
Gunzburger, M.D., Peterson, J.S., Shadid, J.N.: Reducer-order modeling of time-dependent PDEs, with multiple parameters in the boundary data. Comput. Methods Appl. Mech. Engrg. 196, 1030–1047 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
30.
Helenbrook, B.: Mesh deformation using the biharmonic operator. Int. J. Numer. Methods Engrg. 56(7), 1007–1021 (2003)zbMATHCrossRefGoogle Scholar
31.
Ito, K., Ravindran, S.: A reduced-order method for simulation and control of fluid flows. J. Comput. Phys. 143(2), 403–425 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
32.
Kay, D., Loghin, D., Wathen, A.: A preconditioner for the steady-state Navier–Stokes equations. SIAM J. Sci Comput. 24(1), 237–256 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
33.
Knezevic, D.J., Nguyen, N.-C., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for the parametrized unsteady Boussinesq equations. Math. Mod. Meth. Appl. Sci. 21(07), 1415–1442 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
34.
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)MathSciNetzbMATHCrossRefGoogle Scholar
35.
Lancellotti, R.M., Vergara, C., Valdettaro, L., Bose, S., Quarteroni, A.: Large eddy simulations for blood dynamics in realistic stenotic carotids. Int. J. Numer. Methods Biomed. Engng. 33(11), e2868 (2017)MathSciNetCrossRefGoogle Scholar
36.
Maday, Y., Nguyen, N.C., Patera, A.T., Pau, S.H.: A general, multipurpose interpolation procedure: the magic points. Commun. Pur. Appl. Anal. 8(1), 383–404 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
37.
Manzoni, A.: An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier-Stokes flows. ESAIM Math. Modell. Numer. Anal. 48(4), 1199–1226 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
38.
Manzoni, A., Negri, F.: Efficient reduction of PDEs defined on domains with variable shape. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds.) Model Reduction of Parametrized Systems, vol. 17, pp 183–199. Springer, Cham (2017)zbMATHCrossRefGoogle Scholar
39.
Manzoni, A., Quarteroni, A., Rozza, G.: Model reduction techniques for fast blood flow simulation in parametrized geometries. Int. J. Numer Methods Biomed. Engng. 28(6-7), 604–625 (2012)MathSciNetCrossRefGoogle Scholar
40.
Manzoni, A., Quarteroni, A., Rozza, G.: Shape optimization for viscous flows by reduced basis methods and free-form deformation. Int. J. Numer. Methods Fluids 70(5), 646–670 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
41.
May, D.A., Moresi, L.: Preconditioned iterative methods for Stokes flow problems arising in computational geodynamics. Phys. Earth Planet. In. 171(1), 33–47 (2008)CrossRefGoogle Scholar
42.
Negri, F.: Efficient reduction techniques for the simulation and optimization of parametrized systems. Phd thesis EPFL (2015)Google Scholar
43.
Negri, F., Manzoni, A., Amsallem, D.: Efficient model reduction of parametrized systems by matrix discrete empirical interpolation. J. Comput. Phys. 303, 431–454 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
44.
Negri, F., Manzoni, A., Rozza, G.: Reduced basis approximation of parametrized optimal flow control problems for the S,tokes equations. Comput. Math. Appl. 69, 319–336 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
45.
Pagani, S., Manzoni, A., Quarteroni, A.: Numerical approximation of parametrized problems in cardiac electrophysiology by a local reduced basis method. Comput. Meth. Appl. Mech. Engrg. 340, 530–558 (2018)MathSciNetCrossRefGoogle Scholar
46.
Quarteroni, A., Manzoni, A., Negri, F.: Reduced basis methods for partial differential equations: an introduction. Springer, Berlin (2016)zbMATHCrossRefGoogle Scholar
47.
Quarteroni, A., Rozza, G.: Numerical solution of parametrized Navier–Stokes equations by reduced basis methods. Numer. Meth. Part. D.E. 23(4), 923–948 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
48.
Rozza, G., Huynh, D., Manzoni, A.: Reduced basis approximation and error bounds for Stokes flows in parametrized geometries: roles of the inf–sup stability constants. Numer. Math. 125(1), 115–152 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
49.
Saad, Y.: A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14(2), 461–469 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
50.
Segal, A., ur Rehman, M., Vuik, K.: Preconditioners for incompressible Navier-Stokes solvers. Numer. Math. Theory Methods Appl. 3(3), 245–275 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
51.
Semaan, R., Kumar, P., Burnazzi, M., Tissot, G., Cordier, L., Noack, B.: Reduced-order modelling of the flow around a high-lift configuration with unsteady coanda blowing. J Fluid Mech. 800, 72–110 (2016)MathSciNetCrossRefGoogle Scholar
52.
Sieger, D., Botsch, M., Menzel, S.: On shape deformation techniques for simulation-based design optimization. In: Perotto, S., Formaggia, L. (eds.) New Challenges in Grid Generation and Adaptivity for Scientific Computing, volume 5 of SEMA SIMAI Springer Series, pp 281–303. Springer International Publishing, Switzerland (2015)zbMATHCrossRefGoogle Scholar
53.
Silvester, D., Elman, H.C., Kay, D., Wathen, A.: Efficient preconditioning of the linearized Navier–Stokes equations for incompressible flow. J. Comput. Appl Math. 128(1), 261–279 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
54.
Slager, C., Wentzel, J., Gijsen, F., Thury, A., Van der Wal, A., Schaar, J., Serruys, P.: The role of shear stress in the destabilization of vulnerable plaques and related therapeutic implications. Nature Reviews Cardiology 2(9), 456 (2005)Google Scholar
55.
Staten, M.L., Owen, S.J., Shontz, S.M., Salinger, A.G., Coffey, T.S.: A comparison of mesh morphing methods for 3d shape optimization. In: Proceedings of the 20th international meshing roundtable, pp. 293–311. Springer (2011)Google Scholar
56.
Stein, K., Tezduyar, T., Benney, R.: Mesh moving techniques for fluid-structure interactions with large displacements. J. Appl. Mech. 70(1), 58–63 (2003)zbMATHCrossRefGoogle Scholar
57.
Stein, K., Tezduyar, T.E., Benney, R.: Automatic mesh update with the solid-extension mesh moving technique. Comput. Methods Appl. Mech. Engrg. 193 (21–22), 2019–2032 (2004)zbMATHCrossRefGoogle Scholar
58.
Tezduyar, T., Behr, M., Mittal, S., Johnson, A.: Computation of unsteady incompressible flows with the stabilized finite element methods: space-time formulations, iterative strategies and massively parallel implementations. In: New Methods in Transient Analysis, volume 246/AMD, pp. 7–24. ASME, New York (1992)Google Scholar
59.
Turek, S.: Efficient Solvers for Incompressible Flow Problems: An Algorithmic and Computational Approache, vol. 6. Springer Science & Business Media (1999)Google Scholar
60.
Veroy, K., Patera, A.: Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds. Int. J. Numer. Method Fluids 47(8–9), 773–788 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
61.
Vuik, K., Saghir, A., Boerstoel, G.: The Krylov accelerated SIMPLE (R) method for flow problems in industrial furnaces. Int. J. Numer. Methods Fluids 33 (7), 1027–1040 (2000)zbMATHCrossRefGoogle Scholar
62.
Washabaugh, K., Zahr, M., Farhat, C.: On the use of discrete nonlinear reduced-order models for the prediction of steady-state flows past parametrically deformed complex geometries. In: 54th AIAA Aerospace Sciences Meeting, pp. 1814 (2016)Google Scholar
63.
Wathen, A., Silvester, D.: Fast iterative solution of stabilised stokes systems. part i Using simple diagonal preconditioners. SIAM J. Numer. Anal. 30(3), 630–649 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
64.
Weller, J., Lombardi, E., Bergmann, M., Iollo, A.: Numerical methods for low-order modeling of fluid flows based on POD. Int. J. Numer. Methods Fluids 63(2), 249–268 (2010)MathSciNetzbMATHGoogle Scholar
65.
Wirtz, D., Sorensen, D., Haasdonk, B.: A posteriori error estimation for DEIM reduced nonlinear dynamical systems. SIAM J. Sci. Comput. 36(2), A311–A338 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
66.
Wittum, G.: Multi-grid methods for Stokes and Navier-Stokes equations. Numer. Math. 54(5), 543–563 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
67.
Wootton, D.M., Ku, D.N.: Fluid mechanics of vascular systems, diseases, and thrombosis. Ann. Rev. Biomed. Eng. 1(1), 299–329 (1999)CrossRefGoogle Scholar
68.
Yano, M.: A space-time petrov–galerkin certified reduced basis method: application to the Boussinesq equations. SIAM J. Sci. Comput. 36(1), A232–A266 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
69.
Zimmermann, R., Vendl, A., Görtz, S.: Reduced-order modeling of steady flows subject to aerodynamic constraints. AIAA J. 52(2), 255–266 (2014)CrossRefGoogle Scholar

Copyright information

Authors and Affiliations

Niccolò Dal Santo
- 1
Andrea Manzoni
- 2
Email authorView author's OrcID profile

1.École Polytechnique Fédérale de Lausanne (EPFL)School of MathematicsLausanneSwitzerland
2.Department of Mathematics, MOX-Laboratory for Modeling and Scientific ComputingPolitecnico di MilanoMilanoItaly

Hyper-reduced order models for parametrized unsteady Navier-Stokes equations on domains with variable shape

Abstract

Keywords

Mathematics Subject Classification (2010)

Preview

Notes

Acknowledgments

Funding information

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite article