Fluid-Structure Interaction

Fluid-Structure Interaction#

Fluid-Structure Interaction (FSI) involves the coupled interaction between a fluid and a structure, where the motion of one influences the behavior of the other. The Finite Element Method (FEM) is a powerful tool for simulating FSI problems. This section provides an overview of the mathematical formulation for fluid-structure interaction using finite elements.

Coupled Fluid-Structure Equations#

The governing equations for fluid-structure interaction involve the Navier-Stokes equations for the fluid and the equations of motion for the structure. In a partitioned approach, the coupled system is given by:

\[\begin{split} \begin{align} \text{Fluid Domain:} \quad \rho_f \left(\frac{\partial \mathbf{u}_f}{\partial t} + (\mathbf{u}_f \cdot \nabla)\mathbf{u}_f\right) &= -\nabla p_f + \mu_f \nabla^2 \mathbf{u}_f + \mathbf{f}_f + \mathbf{F}_s, \label{eq:fsi_fluid_momentum} \\ \nabla \cdot \mathbf{u}_f &= 0, \label{eq:fsi_fluid_continuity} \\ \text{Structure Domain:} \quad \rho_s \frac{\partial^2 \mathbf{u}_s}{\partial t^2} &= \nabla \cdot \mathbf{P}_s + \mathbf{f}_s, \label{eq:fsi_structure_motion} \end{align} \end{split}\]

where:

  • \(\mathbf{u}_f\) is the fluid velocity,

  • \(p_f\) is the fluid pressure,

  • \(\rho_f\) is the fluid density,

  • \(\mu_f\) is the fluid dynamic viscosity,

  • \(\mathbf{f}_f\) is the fluid body force,

  • \(\mathbf{F}_s\) is the fluid-structure interaction force on the fluid by the structure,

  • \(\mathbf{u}_s\) is the structure displacement,

  • \(\rho_s\) is the structure density,

  • \(\mathbf{P}_s\) is the structure internal force,

  • \(\mathbf{f}_s\) is the structure external force.

Coupling Conditions#

For a fluid-structure interaction problem, coupling conditions at the fluid-structure interface are essential. These conditions ensure continuity of velocities, pressures, and forces between the fluid and structure. A common approach is to enforce kinematic and dynamic conditions:

\[\begin{split} \begin{align} \text{Kinematic Condition:} \quad \mathbf{u}_f &= \mathbf{u}_s, \label{eq:fsi_kinematic_condition} \\ \text{Dynamic Condition:} \quad \mathbf{P}_f \cdot \mathbf{n} &= \mathbf{P}_s \cdot \mathbf{n}, \label{eq:fsi_dynamic_condition} \end{align} \end{split}\]

where \(\mathbf{n}\) is the unit outward normal vector at the fluid-structure interface, and \(\mathbf{P}_f\) and \(\mathbf{P}_s\) are the fluid and structure stresses, respectively.

Weak Formulation#

The weak form of the coupled fluid-structure problem involves the variational formulation of the fluid and structure equations, incorporating the coupling conditions. For the fluid domain, the weak form is given by:

\[ \begin{equation} \int_{\Omega_f} \rho_f \left(\frac{\partial \mathbf{u}_f}{\partial t} + (\mathbf{u}_f \cdot \nabla)\mathbf{u}_f\right) \cdot \mathbf{v}_f \, d\Omega_f + \int_{\Omega_f} \mu_f \nabla \mathbf{u}_f : \nabla \mathbf{v}_f \, d\Omega_f = \int_{\Omega_f} (\mathbf{f}_f + \mathbf{F}_s) \cdot \mathbf{v}_f \, d\Omega_f, \end{equation} \]

where \(\mathbf{v}_f\) is a test function belonging to the function space \(H^1(\Omega_f)\), and \(\Omega_f\) is the fluid domain.

For the structure domain, the weak form is given by:

\[ \begin{equation} \int_{\Omega_s} \rho_s \frac{\partial^2 \mathbf{u}_s}{\partial t^2} \cdot \mathbf{v}_s \, d\Omega_s + \int_{\Omega_s} \nabla \cdot \mathbf{P}_s \cdot \mathbf{v}_s \, d\Omega_s = \int_{\Omega_s} \mathbf{f}_s \cdot \mathbf{v}_s \, d\Omega_s, \end{equation} \]

where \(\mathbf{v}_s\) is a test function belonging to the function space \(H^1(\Omega_s)\), and \(\Omega_s\) is the structure domain.

The coupling conditions, such as \eqref{eq:fsi_kinematic_condition} and \eqref{eq:fsi_dynamic_condition}, should be enforced during the assembly of the global matrices.

References#

The following references provide comprehensive coverage of the mathematical background for fluid-structure interaction using finite elements: [] [] []

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