\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

A reduced basis method by means of transport maps for a fluid–structure interaction problem with slowly decaying Kolmogorov $ n $-width

  • *Corresponding author: Gianluigi Rozza

    *Corresponding author: Gianluigi Rozza 
Abstract Full Text(HTML) Figure(17) / Table(5) Related Papers Cited by
  • The aim of this work is to present a Model Order Reduction (MOR) procedure that is carried out by means of a preprocessing of the snapshots in the offline phase, and to apply it to a Fluid–Structure Interaction (FSI) problem of interest, where the physical domain is two dimensional, the fluid is Newtonian and laminar, and the solid is one dimensional, linear and elastic. This problem exhibits a slow decay of the Kolmogorov $ n $-width: this is reflected, at the numerical level, by a slow decay in the magnitude of the eigenvalues returned by a Proper Orthogonal Decomposition on the solution manifold. By means of a preprocessing procedure, we show how we are able to control the decay of the Kolmogorov $ n $–width of the obtained solution manifold. The preprocessing employed in the manuscript is based on the composition of the snapshots with a map belonging to a family of smooth and invertible mappings from the physical domain into itself. In order to assess the capabilities and the performance of the proposed MOR strategy, we draw a comparison between the results of the novel offline stage and the standard one, as well as a comparison between the novel online phase and the standard one.

    Mathematics Subject Classification: Primary: 65M22, 65M60; Secondary: 76D05, 74F10.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Difference between the current configuration (left) at time $ t $, with the physical domain in a deformed state, and the reference configuration (right) with the physical domain in a resting position

    Figure 2.  Triangular mesh for the FSI test case: mesh size $ h = 0.022 $, for a total of $ 4288 $ vertices. A zoom on a portion of the discretized physical domain is provided for visualization purposes

    Figure 3.  Fluid pressure behavior. The Finite Element solution is represented at time $ t = 0.001 $s, $ t = 0.005 $s and at the final time step $ t = 0.015 $s

    Figure 4.  Fluid velocity behavior. The Finite Element solution is represented at time $ t = 0.001 $s, $ t = 0.005 $s and $ t = 0.015 $s

    Figure 5.  Decay of the eigenvalues $ \lambda_k $ for the POD on the fluid pressure: each $ \lambda_k $ has been normalized by the greatest eigenvalue $ \lambda_0 $

    Figure 6.  Decay of the eigenvalues $ \lambda_k $ for the POD on the fluid velocity: each $ \lambda_k $ has been normalized by the greatest eigenvalue $ \lambda_0 $

    Figure 7.  Peak interpolation. Here we use $ Q = 11 $ sampling points

    Figure 8.  Fluid pressure: the Finite Element solution is represented at time $ t = 0.001 $s, $ t = 0.005 $s and at the final time step $ t = 0.015 $s

    Figure 9.  Fluid pressure: the snapshots are depicted, after transformation, at time $ t = 0.001 $s, $ t = 0.005 $s and $ t = 0.015 $s

    Figure 10.  Decay of the eigenvalues $ \lambda_k $, normalized by the largest one $ \lambda_0 $, before (violet) and after (black) the preprocessing

    Figure 11.  Energy retained by the modes returned by the POD without (violet) and with (black) the preprocessing

    Figure 12.  Pressure snapshots (left column) at time $ t = 0.0026 $ (top) and final time $ t = 0.015 $ (bottom). Pressure approximation obtained with the nonlinear MOR (right column) at time $ t = 0.0026 $ and time $ t = 0.015 $. The reduced simulation has been obtained with $ N = 4 $ basis functions, for each component of the solution of the FSI problem

    Figure 13.  Fluid velocity snapshots (left column) at time $ t = 0.0026 $ (top) and final time $ t = 0.015 $ (bottom). Velocity approximation obtained with the nonlinear MOR (right column) at time $ t = 0.0026 $ and time $ t = 0.015 $. The reduced simulation has been obtained with $ N = 4 $ basis functions, for each component of the solution of the FSI problem

    Figure 14.  Spatial distribution of the approximation error for the fluid pressure (top) and the fluid velocity (bottom), at the final time $ t = 0.015 $s

    Figure 15.  Relative approximation error behavior as a function of time, for the fluid velocity $ \mathit{\boldsymbol{u}}_f $

    Figure 16.  Relative approximation error behavior as a function of time, for the fluid pressure $ p_f $

    Figure 17.  Relative approximation error behavior as a function of time, for the displacement of the compliant wall $ d_s $

    Table 1.  Values of the constants used for the numerical simulation of the FSI test case. Here $ \mathbb{1} $ denotes the indicator function: $ \mathbb{1}(t) = 1 $ if $ t<0.0025 $, $ \mathbb{1}(t) = 0 $ otherwise

    Constant Value Constant Value
    $ R $ $ 6 $ cm $ h_f $ $ 0.5 $cm
    $ \rho_f $ $ 1 $ g/cm$ ^3 $ $ E_s $ $ 0.75\times 10^6 $ dyn/cm$ ^2 $
    $ \nu_f $ $ 0.035 $ Poise $ \nu_s $ $ 0.5 $
    $ b_f $ $ 0 $ $ c_0 $ $ \frac{h_sE_s}{2(1+\nu_s)} $
    $ \rho_s $ $ 1.1 $ g/cm$ ^3 $ $ c_1 $ $ \frac{h_sE_s}{h_f^2(1-\nu_s^2)} $
    $ h_s $ $ 0.1 $ cm $ T_{in} $ $ 2.5\times10^{-3} $
    $ \Delta T $ $ 10^{-4} $ $ p_{in}(t) $ $ 10^3\times[1-\text{cos}(\frac{2\pi t}{T_{in}})] \mathbb{1}_{t<0.0025} $
    T $ 0.015 $ s $ p_{out}(t) $ $ 0 $
     | Show Table
    DownLoad: CSV

    Table 2.  MOR comparison for average approximation error on $ \mathit{\boldsymbol{u}}_f $

    $ N=4 $ $ N=10 $ $ N=20 $
    nonlinear MOR $ 0.039 $ $ 0.026 $ $ 0.029 $
    linear MOR $ 0.59 $ $ 0.078 $ $ 0.00031 $
     | Show Table
    DownLoad: CSV

    Table 3.  MOR comparison for average approximation error on $ p_f $

    $ N=4 $ $ N=10 $ $ N=20 $
    nonlinear MOR $ 0.031 $ $ 0.018 $ $ 0.026 $
    linear MOR $ 0.53 $ $ 0.09 $ $ 0.00036 $
     | Show Table
    DownLoad: CSV

    Table 4.  MOR comparison for average approximation error on $ d_s $

    $ N=4 $ $ N=10 $ $ N=20 $
    nonlinear MOR $ 0.028 $ $ 0.018 $ $ 0.018 $
    linear MOR $ 0.41 $ $ 0.06 $ $ 0.00017 $
     | Show Table
    DownLoad: CSV

    Table 5.  Computational details for the online phase

    Nonlinear MOR $ N=4 $ linear MOR $ N=4 $ linear MOR $ N=20 $
    average number of iterations of Newton's method 3 4 3
    CPU time to solve the online system for one time iteration $ 33 $s $ 11 $s $ 24 $s
     | Show Table
    DownLoad: CSV
  • [1] RBniCS - Reduced order modelling in FEniCS, Available from: http://mathlab.sissa.it/rbnics.
    [2] Multiphenics - Easy prototyping of multiphysics problems in FEniCS, Available from: http://mathlab.sissa.it/multiphenics.
    [3] R. AbgrallD. Amsallem and R. Crisovan, Robust model reduction by ${L}^1$-norm minimization and approximation via dictionaries: Application to nonlinear hyperbolic problems, Advanced Modeling and Simulation in Engineering Sciences, 3 (2016), 1. 
    [4] N. Cagniart, R. Crisovan, Y. Maday and R. Abgrall, Model order reduction for hyperbolic problems: A new framework, preprint, (2017), hal: 01583224.
    [5] F. BallarinA. ManzoniA. Quarteroni and G. Rozza, Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations, International Journal for Numerical Methods in Engineering, 102 (2015), 1136-1161.  doi: 10.1002/nme.4772.
    [6] F. Ballarin and G. Rozza, POD–Galerkin monolithic reduced order models for parametrized fluid-structure interaction problems, International Journal for Numerical Methods in Fluids, 82 (2016), 1010-1034.  doi: 10.1002/fld.4252.
    [7] F. BallarinG. Rozza and Y. Maday, Reduced-order semi-implicit schemes for fluid-structure interaction problems, Model Reduction of Parametrized Systems, MS & A. Model. Simul. Appl., Springer, Cham, 17 (2017), 149-167. 
    [8] F. BernardA. Iollo and S. Riffaud, Reduced-order model for the BGK equation based on POD and optimal transport, Journal of Computational Physics, 373 (2018), 545-570.  doi: 10.1016/j.jcp.2018.07.001.
    [9] L. Bertagna and A. Veneziani, A model reduction approach for the variational estimation of vascular compliance by solving an inverse fluid–structure interaction problem, Inverse Problems, 30, (2014), 055006, 23 pp. doi: 10.1088/0266-5611/30/5/055006.
    [10] W.-J. Beyn and V. Thümmler, Freezing solutions of equivariant evolution equations, SIAM Journal on Applied Dynamical Systems, 3 (2004), 85-116.  doi: 10.1137/030600515.
    [11] N. Cagniart, A Few Non Linear Approaches in Model Order Reduction, Ph.D thesis, École Doctorale De Sciences Mathématiques De Paris Centre, 2018.
    [12] N. CagniartY. Maday and B. Stamm, Model order reduction for problems with large convection effects, Contributions to Partial Differential Equations and Applications, 47 (2019), 131-150. 
    [13] K. Carlberg, Adaptive $h$-refinement for reduced-order models, International Journal for Numerical Methods in Engineering, 102 (2015), 1192-1210.  doi: 10.1002/nme.4800.
    [14] A. Cohen and R. DeVore, Kolmogorov widths under holomorfic mappings, IMA Journal of Numerical Analysis, 36 (2015), 1-12.  doi: 10.1093/imanum/dru066.
    [15] C. M. Colciago and S. Deparis, Reduced order models for fluid-structure interaction problems with applications in haemodynamics, preprint, (2018), arXiv: 1801.06127.
    [16] V. EhrlacherD. LombardiO. Mula and F. Vialard, Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces, ESAIM: M2AN, 54 (2020), 2159-2197.  doi: 10.1051/m2an/2020013.
    [17] A. Ferrero, T. Taddei and L. Zhang, Registration-based model reduction of parameterized two-dimensional conservation laws, Journal of Computational Physics, 457 (2022), 111068, 22 pp. doi: 10.1016/j.jcp.2022.111068.
    [18] J.-F. GerbeauD. Lombardi and E. Schenone, Reduced order model in cardiac electrophysiology with approximated Lax Pairs, Advances in Computational Mathematics, 41 (2015), 1103-1130.  doi: 10.1007/s10444-014-9393-9.
    [19] J. F. Gerbeau and L. Damiano, Approximated Lax Pairs for the reduced order integration of nonlinear evolution equations, Journal of Computational Physics, 265 (2014), 246-269.  doi: 10.1016/j.jcp.2014.01.047.
    [20] C. Greif and K. Urban, Decay of the Kolmogorov N-width for wave problems, Applied Mathematics Letters, 96 (2019), 216-222.  doi: 10.1016/j.aml.2019.05.013.
    [21] M. E. Gurtin, An Introduction to Continuum Mechanics, Mathematics in Science and Engeneering, Academic Press, 1982.
    [22] B. Haasdonk, Reduced basis methods for parametrized PDEs – A tutorial introduction for stationary and instationary problems, Model Reduction and Approximation 2017, Siam, (2017), 65-136.
    [23] B. Haasdonk and M. Ohlberger, Reduced basis method for finite volume approximations of parametrized linear evolution equations, M2AN Math. Model. Numer. Anal., 42 (2008), 277-302.  doi: 10.1051/m2an:2008001.
    [24] J. S. Hesthaven, G. Rozza and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, SpringerBriefs in Mathematics. BCAM SpringerBriefs. Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2016. doi: 10.1007/978-3-319-22470-1.
    [25] A. Iollo and D. Lombardi, Advection modes by optimal mass transfer, Phys. Rev. E, 89 (2014), 022923. 
    [26] E. N. KaratzasF. Ballarin and G. Rozza, Projection-based reduced order models for a cut finite element method in parametrized domains, Comput. Math. Appl., 79 (2020), 833-851.  doi: 10.1016/j.camwa.2019.08.003.
    [27] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM Journal on Numerical Analysis, 40 (2002), 492-515.  doi: 10.1137/S0036142900382612.
    [28] T. LassilaA. ManzoniA. Quarteroni and G. Rozza, Model order reduction in fluid dynamics: Challenges and perspectives, Reduced Order Methods for Modeling and Computational Reduction, Springer International Publishing, 9 (2014), 235-273.  doi: 10.1007/978-3-319-02090-7_9.
    [29] T. Lassila, A. Quarteroni and G. Rozza, A reduced basis model with parametric coupling for fluid-structure interaction problems, SIAM Journal on Scientific Computing, 34 (2012), A1187-A1213. doi: 10.1137/110819950.
    [30] K. Lee and K. T. Carlberg, Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders, Journal of Computational Physics, 404 (2020), 108973, 32 pp. doi: 10.1016/j.jcp.2019.108973.
    [31] A. Logg, K. A. Mardal and G. Wells, Automated Solution of Differential Equations by the Finite Element Method, Springer-Verlag, Berlin, 2012.
    [32] J. M. Melenk, On n-widths for elliptic problems, Journal of Mathematical Analysis and Applications, 247 (2000), 272-289.  doi: 10.1006/jmaa.2000.6862.
    [33] S. Sy and C. Murea, Algorithm for solving fluid–structure interaction problem on a global moving mesh, Coupled Systems Mechanics, 1 (2012).
    [34] C. M. Murea and S. Sy, A fast method for solving fluid–structure interaction problems numerically, International Journal for Numerical Methods in Fluids, 60 (2009), 1149-1172.  doi: 10.1002/fld.1931.
    [35] N. J. Nair and M. Balajewicz, Transported snapshot model order reduction approach for parametric, steady-state fluid flows containing parameter-dependent shocks, International Journal for Numerical Methods in Engineering, 117 (2019), 1234-1262.  doi: 10.1002/nme.5998.
    [36] N.-C. NguyenG. Rozza and A. T. Patera, Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers' equation, Calcolo, 46 (2009), 157-185.  doi: 10.1007/s10092-009-0005-x.
    [37] M. NoninoF. Ballarin and G. Rozza, A monolithic and a partitioned, reduced basis method for fluid-structure interaction problems, Fluids, 6 (2021), 229. 
    [38] M. Ohlberger and S. Rave, Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing, Comptes Rendus Mathematique, 351 (2013), 901-906.  doi: 10.1016/j.crma.2013.10.028.
    [39] M. Ohlberger and S. Rave, Reduced basis methods: Success, limitations and future challenges, Proceedings of the Conference Algoritmy, (2016), 1-12.
    [40] B. Peherstorfer, Model reduction for transport-dominated problems via online adaptive bases and adaptive sampling, SIAM Journal on Scientific Computing, 42 (2020), A2803-A2836. doi: 10.1137/19M1257275.
    [41] A. Quarteroni and L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system, Handbook of Numerical Analysis, 12 (2004), 3-127. 
    [42] A. QuarteroniM. Tuveri and A. Veneziani, Computational vascular fluid dynamics: Problems, models and methods, Computing and Visualization in Science, 2 (2000), 163-197. 
    [43] J. Reiss, P. Schulze, J. Sesterhenn and V. Mehrmann, The shifted proper orthogonal decomposition: A mode decomposition for multiple transport phenomena, SIAM Journal on Scientific Computing, 40 (2018), A1322-A1344. doi: 10.1137/17M1140571.
    [44] T. Richter, Fluid–Structure Interactions. Model, Analysis and Finite Element, Lecture Notes in Computational Science and Engineering, Springer International Publishing, 2017.
    [45] C. W. RowleyI. G. KevrekidisJ. E. Marsden and K. Lust, Reduction and reconstruction for self-similar dynamical systems, Nonlinearity, 16 (2003), 1257-1275.  doi: 10.1088/0951-7715/16/4/304.
    [46] G. RozzaD. B. P. Huynh and A. T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Arch. Comput. Meth. Eng., 15 (2008), 229-275.  doi: 10.1007/s11831-008-9019-9.
    [47] T. Taddei and L. Zhang, Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs, ESAIM: Mathematical Modelling and Numerical Analysis, 55 (2021), 99-130.  doi: 10.1051/m2an/2020073.
    [48] T. Taddei and L. Zhang, Registration-based model reduction in complex two-dimensional geometries, Journal of Scientific Computing, 88 (2021), Paper No. 79, 25 pp. doi: 10.1007/s10915-021-01584-y.
    [49] D. Torlo, Model reduction for advection dominated hyperbolic problems in an ALE framework: Offline and online phases, preprint, (2020), arXiv: 2003.13735.
    [50] G. Welper, H and HP-adaptive interpolation by transformed snapshots for parametric and stochastic hyperbolic PDEs, preprint, (2017), arXiv: 1710.11481.
    [51] G. Welper, Interpolation of functions with parameter dependent jumps by transformed snapshots, SIAM Journal on Scientific Computing, 39 (2017), A1225-A1250. doi: 10.1137/16M1059904.
    [52] G. Welper, Transformed snapshot interpolation with high resolution transforms, SIAM Journal on Scientific Computing, 42 (2020), A2037-A2061. doi: 10.1137/19M126356X.
  • 加载中

Figures(17)

Tables(5)

SHARE

Article Metrics

HTML views(432) PDF downloads(206) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return