Finite element approximation for a reformulation of a 3D fluid-2D plate interaction system

TL;DR

This work develops a finite element framework for a 3D viscous fluid interacting with a 2D Kirchhoff plate, achieved by reformulating the plate’s fourth-order equation into a coupled second-order system using an auxiliary variable. A Lagrange multiplier enforces the interface coupling via the trace of the mean-zero fluid pressure, producing a saddle-point formulation that supports a partitioned, fixed-point solution. The authors prove well-posedness and stability for time-discrete and fully discrete schemes and derive a priori error estimates with convergence rates driven by the time step and spatial mesh sizes, validating the method with numerical experiments that confirm optimal rates and apply the approach to a physically meaningful free-vibrating plate. The approach offers flexibility in spatial discretization for the plate and demonstrates scalability through domain-decomposition ideas and a solver strategy that leverages parallel sparse LU decomposition (MUMPS).

Abstract

We study a finite element approximation of a coupled fluid-structure interaction consisting of a three-dimensional incompressible viscous fluid governed by the unsteady Stokes equations and a two-dimensional elastic plate. To avoid the use of $H^2-$conforming or nonconforming $\mathbb{P}_2$-Morley plate elements, the fourth-order plate equation is reformulated into a system of coupled second-order equations using an auxiliary variable. The coupling condition is enforced using a Lagrange multiplier representing the trace of the mean-zero fluid pressure on the interface. We establish well-posedness and stability results for the time-discrete and fully-discrete problems, and derive a priori error estimates. A partitioned domain decomposition algorithm based on a fixed-point iteration is employed for the numerical solution. Numerical experiments verify the theoretical rates of convergence in space and time using manufactured solutions, and demonstrate the applicability of the method to a physical problem.

Figures (4)

• Figure 1: Convergence test: vertical component $u_3$ of the velocity at $z = 0$ (left), velocity of the plate $\dot{w}$ (center), and their absolute difference (right) at $t = 1e-03$.
• Figure 2: The absolute value of the integral $\int_{\Omega_p}\dot{w}_h^{n}\,d\Omega_p$ for different spatial mesh sizes $h_p$.
• Figure 3: Free vibrating plate: maximum displacement of the plate (left), and kinetic energies associated to the fluid and the plate, and the corresponding total kinetic energy of the system (right) over time.
• Figure 4: Free vibrating plate: plate velocity $\dot{w}$ (left), vertical component $u_3$ of the fluid velocity at $z = 0$ (center), and their absolute difference (right) at $t = 0.01 \text{ (first row)}, 0.05 \text{ (second row)}, \text{ and }0.1 \text{ (third row)}$.

Theorems & Definitions (9)

• Lemma 3.1
• proof
• Theorem 3.2
• Theorem 3.3
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof