Category:Linear Hydroelasticity

Problems in Linear Water-Wave theory in which there is an elastic body.

Expansion in Modes

The basic idea is to use the same solution method as for a rigid body) except to include elastic modes. While there will be an infinite number of these modes in general, in practice only a few of the lowest modes will be important unless the body is very flexible.

Finite Element Method

The finite element method is ideally suited to analyse flexible bodies. In the standard FEM notation the dynamic equation of motion in matrix form can be expressed as:

[math]\displaystyle{ \begin{bmatrix}K\end{bmatrix}\begin{bmatrix}D\end{bmatrix}+ \begin{bmatrix}S\end{bmatrix}\begin{bmatrix}\dot D\end{bmatrix}+ \begin{bmatrix}M\end{bmatrix}\begin{bmatrix}\ddot D\end{bmatrix}= \begin{bmatrix}F(t)\end{bmatrix} }[/math]

where

[math]\displaystyle{ \begin{bmatrix}K\end{bmatrix} }[/math] is structural stiffness matrix, [math]\displaystyle{ \begin{bmatrix}S\end{bmatrix} }[/math] is structural damping matrix, [math]\displaystyle{ \begin{bmatrix}M\end{bmatrix} }[/math] is structural mass matrix, [math]\displaystyle{ \begin{bmatrix}D\end{bmatrix} }[/math] is generalized nodal displacements vector, [math]\displaystyle{ \begin{bmatrix}F\end{bmatrix} }[/math] is generalized force vector (fluid forces, gravity forces,...).

Left-hand side of the global FEM matrix equation represents "dry" (in vacuuo) structure, while the right-hand side includes fluid forces (and coupling between the surrounding fluid and the structure).

Frequency Domain Problem

We consider the problem in the Frequency Domain so that [math]\displaystyle{ \begin{bmatrix}D\end{bmatrix}=\begin{bmatrix}w\end{bmatrix}\,e^{-i\omega t} }[/math].

The eigenvalue problem for the "dry" natural vibrations yields:

[math]\displaystyle{ \left( \begin{bmatrix}K\end{bmatrix}-\lambda \begin{bmatrix}M\end{bmatrix} \right )\begin{bmatrix}w\end{bmatrix}=\begin{bmatrix}0\end{bmatrix} }[/math]

As a solution of the eigenvalue problem for each natural mode one obtains [math]\displaystyle{ \omega_n }[/math], the n-th dry natural frequency and [math]\displaystyle{ \begin{bmatrix}w_n\end{bmatrix} }[/math], the corresponding dry natural mode. Note that only the first modes are accurate approximations.

We expand the displacement in [math]\displaystyle{ N }[/math] modes,

[math]\displaystyle{ w = \sum_{\nu} \xi_n \mathbf{w}_{\nu} }[/math]

Associated with each of these modes is a normal motion on the body surface given by [math]\displaystyle{ \mathbf{n}_{\nu} }[/math]. The first modes are the standard rigid modes.

Equations for the Fluid

The equation for the fluid are as follows

[math]\displaystyle{ \begin{align} \Delta\phi &=0, &-h\lt z\lt 0,\,\,\mathbf{x} \in \Omega \\ \partial_z\phi &= 0, &z=-h, \\ \partial_z \phi &= \alpha \phi, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]

(note that the last expression can be obtained from combining the expressions:

[math]\displaystyle{ \begin{align} \partial_z \phi &= -\mathrm{i} \omega \zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \\ \mathrm{i} \omega \phi &= g\zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]

where [math]\displaystyle{ \alpha = \omega^2/g \, }[/math]) The body boundary condition for a radiation mode [math]\displaystyle{ m }[/math] is just

[math]\displaystyle{ \partial_{n}\phi=\mathbf{n}_\nu,\ \ \mathbf{x}\in\partial\Omega_{\mathrm{B}}, }[/math]

where [math]\displaystyle{ \mathbf{n}_{\nu} }[/math] is the normal derivative of the [math]\displaystyle{ \nu }[/math] mode on the body surface directed out of the fluid.

Fluid Structure Equations

We substitute the expansion for the potential into the equations in the frequency domain and we obtain

[math]\displaystyle{ \sum_{\nu} K_{\mu\nu}\zeta_{\nu} -\mathrm{i}\omega\sum_{\nu} S_{\mu\nu}\zeta_{\nu} -\omega^2 \sum_{\nu} M_{\mu\nu}\zeta_{\nu}=-\mathrm{i}\omega\rho\iint_{\partial\Omega_{B}} \left(\phi^{\mathrm{I}} + \phi^{\mathrm{D}} + \sum_{\nu} \zeta_\nu \phi_{\nu}^{\mathrm{R}}\right) \mathbf{n}_{\mu}\, dS - \sum_{\nu} C_{\mu\nu}\xi_{\nu}, }[/math]

We define the added mass matrix by

[math]\displaystyle{ A_{\mu\nu} = \mathrm{Re} \left\{\rho\iint_{\partial\Omega_{B}} \phi_{\nu}^{\mathrm{R}} \mathbf{n}_{\mu}\, dS \right\} }[/math]

and the damping matrix by

[math]\displaystyle{ B_{\mu\nu} = \mathrm{Im} \left\{ \omega \rho\iint_{\partial\Omega_{B}} \phi_{\nu}^{\mathrm{R}} \mathbf{n}_{\mu}\, dS \right\} }[/math]

and the forcing vector by

[math]\displaystyle{ f_{\mu} = \mathrm{i}\omega\rho\iint_{\partial\Omega_{B}} \left(\phi^{\mathrm{I}} + \phi^{\mathrm{D}} \right) \mathbf{n}_{\mu}\, dS }[/math]

Then the equations can be expressed as follows.

[math]\displaystyle{ \left[\mathbf{K} -\mathrm{i}\mathbf{S} -\omega^2 \left(\mathbf{M} + \mathbf{A} \right) + \mathrm{i}\omega \mathbf{B} + \mathbf{C} \right] \vec{\zeta} = \mathbf{f} }[/math]

where [math]\displaystyle{ \mathbf{K} }[/math] is the stiffnes matrix, [math]\displaystyle{ \mathbf{S} }[/math] is the structural damping matrix, [math]\displaystyle{ \mathbf{M} }[/math] is the mass matrix, [math]\displaystyle{ \mathbf{A} }[/math] is the added mass matrix, [math]\displaystyle{ \mathbf{B} }[/math] is the damping matrix, [math]\displaystyle{ \mathbf{C} }[/math] is the hydrostatic matrix, [math]\displaystyle{ \vec{\zeta} }[/math] is the vector of body displacements and [math]\displaystyle{ \mathbf{f} }[/math] is the force.