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

TL;DR

This paper develops and analyzes a finite element method for solving a coupled elasto-acoustic eigenproblem using a Herrmann pressure formulation, providing convergence proofs, error estimates, and numerical validation in 2D and 3D.

Contribution

It introduces a non conforming, locking-free finite element approach for a coupled fluid-structure eigenproblem with variable coefficients, including error analysis and adaptive estimation.

Findings

Proves convergence and error bounds for the proposed method.

Demonstrates the method's effectiveness through numerical tests in 2D and 3D.

Provides an efficient a posteriori error estimator for the coupled problem.

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.