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Finite element analysis for a Herrmann pressure formulation of the elastoacoustic problem with variable coefficients

Arbaz Khan, Felipe Lepe, David Mora, Ricardo Ruíz-Baier, Jesus Vellojin

TL;DR

This work formulates and analyzes a coupled elasto-acoustic eigenproblem using a Herrmann pressure variable in the solid and a displacement-based fluid model. A locking-free, nonconforming finite-element method is developed with inf-sup stable solids (MINI or Taylor–Hood) and BD_M fluids, augmented by a corrected interface interpolation to preserve accuracy at the fluid–solid boundary. The authors establish spectral convergence for a non-compact problem, derive a residual-based a posteriori error estimator with proven reliability and efficiency (via Helmholtz decomposition and weighted norms), and demonstrate the approach through 2D and 3D numerical experiments including adaptive refinement. The results show robust convergence, absence of spurious eigenvalues, and effective adaptive refinement in geometries with singularities, highlighting practical utility for sloshing and elasto-acoustic vibration analyses with variable coefficients.

Abstract

In two and three dimensions, this study is focused on the numerical analysis of an eigenproblem associated with a fluid-structure model for sloshing and elasto-acoustic vibration. We use a displacement-Herrmann pressure formulation for the solid, while for the fluid, a pure displacement formulation is considered. Under this approach we propose a non conforming locking-free method based on classic finite elements to approximate the natural frequencies (of the eigenmodes) of the coupled system. Employing the theory for non-compact operators we prove convergence and error estimates. Also we propose an a posteriori error estimator for this coupled problem which is shown to be efficient and reliable. All the presented theory is contrasted with a set of numerical tests in 2D and 3D.

Finite element analysis for a Herrmann pressure formulation of the elastoacoustic problem with variable coefficients

TL;DR

This work formulates and analyzes a coupled elasto-acoustic eigenproblem using a Herrmann pressure variable in the solid and a displacement-based fluid model. A locking-free, nonconforming finite-element method is developed with inf-sup stable solids (MINI or Taylor–Hood) and BD_M fluids, augmented by a corrected interface interpolation to preserve accuracy at the fluid–solid boundary. The authors establish spectral convergence for a non-compact problem, derive a residual-based a posteriori error estimator with proven reliability and efficiency (via Helmholtz decomposition and weighted norms), and demonstrate the approach through 2D and 3D numerical experiments including adaptive refinement. The results show robust convergence, absence of spurious eigenvalues, and effective adaptive refinement in geometries with singularities, highlighting practical utility for sloshing and elasto-acoustic vibration analyses with variable coefficients.

Abstract

In two and three dimensions, this study is focused on the numerical analysis of an eigenproblem associated with a fluid-structure model for sloshing and elasto-acoustic vibration. We use a displacement-Herrmann pressure formulation for the solid, while for the fluid, a pure displacement formulation is considered. Under this approach we propose a non conforming locking-free method based on classic finite elements to approximate the natural frequencies (of the eigenmodes) of the coupled system. Employing the theory for non-compact operators we prove convergence and error estimates. Also we propose an a posteriori error estimator for this coupled problem which is shown to be efficient and reliable. All the presented theory is contrasted with a set of numerical tests in 2D and 3D.

Paper Structure

This paper contains 16 sections, 17 theorems, 106 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model problem
  3. Numerical discretization
  4. The interface approximation
  5. The discrete eigenvalue problem
  6. Spectral approximation
  7. A posteriori error analysis
  8. Residual based a posteriori error estimator
  9. Reliability
  10. Efficiency
  11. Numerical experiments
  12. Accuracy test
  13. Fluid-structure interaction in a half-filled barrel
  14. A posteriori error estimation
  15. 2D example
  16. ...and 1 more sections

Key Result

lemma thmcounterlemma

If $((\boldsymbol{f},\boldsymbol{g}),g)\in\mathcal{G}$, then $\mathbf{T}((\boldsymbol{f},\boldsymbol{g}),g):=((\overline{\boldsymbol{u}},\overline{\boldsymbol{w}}),\overline{p})\in\mathcal{G}_{\mathbf{H}}$ is the unique solution of eq:source_cont. Moreover, there exist $\alpha\in(1/2,1]$, $\beta\in

Figures (13)

  • Figure 1: Sketch of a fluid-structure interaction domain with sub-boundaries. Here, $\Omega_S$ and $\Omega_F$ denote the solid and fluid subdomains, respectively.
  • Figure 1: Test \ref{['subsec:accuracy-test']}. Schematic of 2D computational domains indicating dimensions, boundaries, subdomains, and using a coarse mesh with $N=2$.
  • Figure 2: Example \ref{['subsec:accuracy-test']}. Comparison between the first fourth lowest order computed elasto-acoustic modes on $\Omega_2$. The solid domain have been warped by a sufficiently large factor in order to observe the deformation.
  • Figure 3: Example \ref{['subsec:3Dbarrel']}. The computational domain of the half filled barrel.
  • Figure 4: Example \ref{['subsec:3Dbarrel']}. Sample numerical solutions of the first four elasto-acoustic modes on the barrel domain. The solid subdomain has been warped by a sufficiently large factor in order to observe the deformation.
  • ...and 8 more figures

Theorems & Definitions (17)

  • lemma thmcounterlemma
  • corollary thmcountercorollary
  • theorem 2.1
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • theorem 4.1
  • ...and 7 more