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Numerical methods for solving the linearized model of a hinged-free reduced plate arising in flow structure interactions

Raj Narayan Dhara, Krzysztof E. Rutkowski, Katarzyna Szulc

TL;DR

This work studies stationary states of a nonlinear, nonlocal hinged-free plate model coupled to flow through a wind parameter $α$, capturing flow-structure interaction and vortex shedding. It casts the problem in a linearized variational framework $a(U,V)+μ(U_x,V_x)=(G,V)+α(U_y,V)$ with $μ=\int_Ω u_x^2−P$, and compares two finite element approaches: a rectangular-grid, separable-variable discretization and a high-order triangular (P4) discretization. The key finding is that increasing $|α|$ increases the number of stationary solutions, indicating a rich, non-self-adjoint stationary structure tied to long-time attractor dynamics. The results suggest that higher-order P4 discretizations better capture the stationary branches, informing stability and flutter analyses in engineering applications of flow-structure interactions.

Abstract

The problem of partially hinged and partially free rectangular plate that aims to represent a suspension bridge subject to some external forces (for example the wind) is considered in order to model and simulate the unstable end behavior. Such a problem can be modeled by a plate evolution equation, which is nonlinear with a nonlocal stretching effect in the spanwise direction. The external forces are periodic in time and cause the vortex shedding on the structure (on the surface of the plate) and thus it may cause damage to the material. Numerical study of the behavior of steady state solutions for different values of the force velocity are provided with two finite element methods of different type.

Numerical methods for solving the linearized model of a hinged-free reduced plate arising in flow structure interactions

TL;DR

This work studies stationary states of a nonlinear, nonlocal hinged-free plate model coupled to flow through a wind parameter , capturing flow-structure interaction and vortex shedding. It casts the problem in a linearized variational framework with , and compares two finite element approaches: a rectangular-grid, separable-variable discretization and a high-order triangular (P4) discretization. The key finding is that increasing increases the number of stationary solutions, indicating a rich, non-self-adjoint stationary structure tied to long-time attractor dynamics. The results suggest that higher-order P4 discretizations better capture the stationary branches, informing stability and flutter analyses in engineering applications of flow-structure interactions.

Abstract

The problem of partially hinged and partially free rectangular plate that aims to represent a suspension bridge subject to some external forces (for example the wind) is considered in order to model and simulate the unstable end behavior. Such a problem can be modeled by a plate evolution equation, which is nonlinear with a nonlocal stretching effect in the spanwise direction. The external forces are periodic in time and cause the vortex shedding on the structure (on the surface of the plate) and thus it may cause damage to the material. Numerical study of the behavior of steady state solutions for different values of the force velocity are provided with two finite element methods of different type.
Paper Structure (6 sections, 2 theorems, 30 equations, 7 figures)

This paper contains 6 sections, 2 theorems, 30 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Stationary case and Linearisation
  4. Numerical analysis
  5. Rectangular grid case
  6. Triangular mesh case

Key Result

Lemma 3.1

Let $S>0$ and $0\leq P<\lambda_1$ be given and suppose that $u$ is the solution to the problem eq:8.1. There exists constant $C>0$ depending on $\alpha$, $g$, $P$ and $S$ such that $u\in B_C(\mathcal{K}^2)$ where $B_C$ is a ball of radius $C$.

Figures (7)

  • Figure 2.1: Rectangular plate representing the domain $\Omega$.
  • Figure 4.1: Uniform Rectangular Mesh on $[0, \pi]\times [-1/2,1/2]$;
  • Figure 4.2: Basis functions for $x$ variable corresponding to 6 interior nodes
  • Figure 4.3: Solution $U_{-125}$ with multiple zeros at the $x$-direction. Case for $\alpha=-125$ .
  • Figure 4.4: Solution $U_{1,-460}$ with no zeros at the $x$-direction. Case $m=1$ with $\alpha=-520\leq -460=\alpha_1$.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Lemma 3.1
  • Lemma 3.2