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Asymptotically proved numerical coupling of a 2D flexural porous plate with the 3D Stokes fluid

Maxime Krier, Julia Orlik, Grigory Panasenko, Konrad Steiner

Abstract

This paper presents an efficient coupling of the 3D Stokes flow interacting with an effective perforated periodic heterogeneous anisotropic 2D plate. The effective model was obtained by the asymptotic analysis in earlier works and here an effective numerical algorithm is given. By $Q_3$ or bi-cubic spacial interpolation the time-dependent problem was reduced to an algebraic system of ordinary differential equation in time. Different examples were given, demonstrating the influence of the structural plate parameters on the solution.

Asymptotically proved numerical coupling of a 2D flexural porous plate with the 3D Stokes fluid

Abstract

This paper presents an efficient coupling of the 3D Stokes flow interacting with an effective perforated periodic heterogeneous anisotropic 2D plate. The effective model was obtained by the asymptotic analysis in earlier works and here an effective numerical algorithm is given. By or bi-cubic spacial interpolation the time-dependent problem was reduced to an algebraic system of ordinary differential equation in time. Different examples were given, demonstrating the influence of the structural plate parameters on the solution.
Paper Structure (11 sections, 10 theorems, 83 equations, 26 figures, 1 table)

This paper contains 11 sections, 10 theorems, 83 equations, 26 figures, 1 table.

Table of Contents

  1. Introduction
  2. Multiscale problem
  3. Macroscopic model description
  4. Computation of the effective textile coefficients
  5. Computation of the permeability
  6. Monolithic FSI solver
  7. Simulation results
  8. Qualitative description of stiffness tensors
  9. Quantitative description of stiffness tensors
  10. Extension with anisotropic model parameters
  11. Conclusions

Key Result

Lemma 3.1

\newlabelproposition:cellProblemsStructure0 For each $i,j\in\{1,2\}$, there exists a unique cell solution $\bm{\chi}_{ij}^{M,B}\in H^1_{\#,0}(Y^s)^3$ to the cell problems eq:cellProblemStructure, respectively.

Figures (26)

  • Figure 1: Example of a reference cell for a twill woven filter. The notation of $S_\varepsilon^c$ has to be updated
  • Figure 1: Illustration of the structure domain $\Omega_\varepsilon^{M,s}$ (left), solid part of unit cell $Y^s$ (center) and quarter of the unit cell $\tilde{Y}^s$ (right) for a plain woven filter.
  • Figure 1: Flow solutions $(\bm{v}_2,p_2)$ (top) and $(\bm{v}_3,p_3)$ (bottom) for a twill woven filter. The remaining flow solution is of similar nature due to symmetry of the structure.
  • Figure 1: Hermite splines on unit interval.
  • Figure 1: Displacement of homogenized textile under applied tension in $x_2$-direction for a zero (left) and a large, non-zero value of $a^\text{hom}_{2211}$ (right). Free lateral boundary left and right. Colors indicate displacement in $x_1$-direction.
  • ...and 21 more figures

Theorems & Definitions (18)

  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 4.1
  • Example 4.2
  • Definition 4.3
  • Proposition 4.4
  • Lemma 6.1
  • Theorem 6.2
  • ...and 8 more