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

Monica Nonino; Francesco Ballarin; Gianluigi Rozza; Yvon Maday

doi:10.3934/acse.2023002

Article Contents

2023, Volume 1, Issue 1: 36-58. Doi: 10.3934/acse.2023002

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

1.
University of Vienna, Faculty of Mathematics, Vienna, Austria
2.
Università Cattolica del Sacro Cuore, Department of Mathematics and Physics, Brescia, Italy
3.
mathLab, Mathematics Area, SISSA, Trieste, Italy
4.
Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions, Paris, France

^*Corresponding author: Gianluigi Rozza
^*Corresponding author: Gianluigi Rozza

Received: November 2022

Revised: February 2023

Early access: March 2023

Published: March 2023

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Abstract

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.

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

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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

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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

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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

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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

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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 $

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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 $

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

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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

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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

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

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

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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

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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

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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

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

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

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

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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 $

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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 $

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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 $

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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 $

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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

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