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A Projection-Based Time-Segmented Reduced Order Model for Fluid-Structure Interactions

Qijia Zhai, Shiquan Zhang, Pengtao Sun, Xiaoping Xie

Abstract

In this paper, a type of novel projection-based, time-segmented reduced order model (ROM) is proposed for dynamic fluid-structure interaction (FSI) problems based upon the arbitrary Lagrangian--Eulerian (ALE)-finite element method (FEM) in a monolithic frame, where spatially, each variable is separated from others in terms of their attribution (fluid/structure), category (velocity/pressure) and component (horizontal/vertical) while temporally, the proper orthogonal decomposition (POD) bases are constructed in some deliberately partitioned time segments tailored through extensive numerical trials. By the combination of spatial and temporal decompositions, the developed ROM approach enables prolonged simulations under prescribed accuracy thresholds. Numerical experiments are carried out to compare numerical performances of the proposed ROM with corresponding full-order model (FOM) by solving a two-dimensional FSI benchmark problem that involves a vibrating elastic beam in the fluid, where the performance of offline ROM on perturbed physical parameters in the online phase is investigated as well. Extensive numerical results demonstrate that the proposed ROM has a comparable accuracy to while much higher efficiency than the FOM. The developed ROM approach is dimension-independent and can be seamlessly extended to solve high dimensional FSI problems.

A Projection-Based Time-Segmented Reduced Order Model for Fluid-Structure Interactions

Abstract

In this paper, a type of novel projection-based, time-segmented reduced order model (ROM) is proposed for dynamic fluid-structure interaction (FSI) problems based upon the arbitrary Lagrangian--Eulerian (ALE)-finite element method (FEM) in a monolithic frame, where spatially, each variable is separated from others in terms of their attribution (fluid/structure), category (velocity/pressure) and component (horizontal/vertical) while temporally, the proper orthogonal decomposition (POD) bases are constructed in some deliberately partitioned time segments tailored through extensive numerical trials. By the combination of spatial and temporal decompositions, the developed ROM approach enables prolonged simulations under prescribed accuracy thresholds. Numerical experiments are carried out to compare numerical performances of the proposed ROM with corresponding full-order model (FOM) by solving a two-dimensional FSI benchmark problem that involves a vibrating elastic beam in the fluid, where the performance of offline ROM on perturbed physical parameters in the online phase is investigated as well. Extensive numerical results demonstrate that the proposed ROM has a comparable accuracy to while much higher efficiency than the FOM. The developed ROM approach is dimension-independent and can be seamlessly extended to solve high dimensional FSI problems.
Paper Structure (13 sections, 1 theorem, 35 equations, 24 figures, 2 tables)

This paper contains 13 sections, 1 theorem, 35 equations, 24 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model description and finite element method for FSI
  3. Model of dynamic FSI problems
  4. ALE mapping
  5. Monolithic ALE-FEM for FSI
  6. Reduced order model for FSI
  7. Offline Phase
  8. Online Phase
  9. Numerical Experiments
  10. Comparisons of FOM and ROM
  11. Motivation of partitioning the spatial and temporal dimension
  12. Conclusion
  13. Acknowledgement

Key Result

Lemma 2.1

Gastaldi2001Nobile;Formaggia1999 For any $t\in(0,T]$, $\bm u_f(\bm x,t)\in H^1(\Omega_f(t))^d$ and $\partial_t^{\mathcal{\hat{\bm A}}_f} {\bm u}_f(\bm x,t)\in H^1(\Omega_f(t))^d$ if and only if $\hat{\bm u}(\hat{\bm x},t)=\bm u(\bm x,t)\circ \mathcal{\hat{\bm A}}_f(\hat{\bm x},t)\in H^1(\hat{\Omega}

Figures (24)

  • Figure 1: A schematic domain of FSI benchmark problem, where the fluid channel $\Omega_f(t)$ is in blue, the red-colored structural domain $\Omega_s(t)$ represents a elastic beam behind a rigid cylinder. From the left to right, lines in purple, cyan, green and orange represent the inlet $\Gamma_{in}$, the fluidic channel wall $\Gamma_{walls}$, the fluid-structure interface $\Gamma_{FSI}(t)$, and the outlet $\Gamma_{out}$, respectively.
  • Figure 2: Computational domain of the FSI benchmark problem
  • Figure 3: The triangular mesh of benchmark domain.
  • Figure 4: Vibration curve (the $y$-displacement trajectory) of point $A$ obtained from the FOM result.
  • Figure 5: Number of POD bases' selections over the time interval [2s,15s].
  • ...and 19 more figures

Theorems & Definitions (1)

  • Lemma 2.1