Three-Dimensional Floating Elastic Plate

Three Dimensional Problem

Equations of Motion

For a classical thin plate, the equation of motion is given by

[math]\displaystyle{ D\nabla ^4 w + \rho _i h w = p }[/math]

Equation ((plate)) is subject to the free edge boundary conditions for a thin plate

[math]\displaystyle{ \frac{\partial ^{2}w}{\partial n^{2}}+\nu \frac{\partial ^{2}w}{\partial s^{2}}=0,\;\;\;=\textrm{and= }\mathrm{\;\;\;}\frac{\partial ^{3}w}{ \partial n^{3}}+\left( 2-\nu \right) \frac{\partial ^{3}w}{\partial n\partial s^{2}}=0, (boundaryplate) }[/math]

Hildebrand65 where [math]\displaystyle{ n }[/math] and [math]\displaystyle{ s }[/math] denote the normal and tangential directions respectively.

The pressure, [math]\displaystyle{ p }[/math], is given by the linearized Bernoulli's equation at the water surface,

[math]\displaystyle{ p=-\rho \frac{\partial \phi }{\partial t}-\rho gW (pressure) }[/math]

where [math]\displaystyle{ \Phi }[/math] is the velocity potential of the water, [math]\displaystyle{ \rho }[/math] is the density of the water, and [math]\displaystyle{ g }[/math] is the acceleration due to gravity.

We now introduce non-dimensional variables. We non-dimensionalise the length variables with respect to [math]\displaystyle{ a }[/math] where the surface area of the floe is [math]\displaystyle{ 4a^{2}. }[/math] We non-dimensionalise the time variables with respect to [math]\displaystyle{ \sqrt{g/a} }[/math] and the mass variables with respect to [math]\displaystyle{ \rho a^{3} }[/math]. In the non-dimensional variables equations ((plate)) and ((pressure) ) become

[math]\displaystyle{ \beta \nabla ^{4}\bar{w}+\gamma \frac{\partial ^{2}\bar{w}}{\partial \bar{t} ^{2}}=\frac{\partial \bar{\Phi}}{\partial \bar{t}}-\bar{w}, (n-d_ice) }[/math]

where

[math]\displaystyle{ \beta =\frac{D}{g\rho a^{4}}\;\;\{mathrm and}\ \ = \gamma =\frac{\rho _{i}h}{\rho a}. }[/math]

We shall refer to [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ \gamma }[/math] as the stiffness and mass respectively.

We will determine the response of the ice floe to wave forcing of a single frequency (the response for more complex wave forcing can be found by superposition of the single frequency solutions). Since the equations of motion are linear the displacement and potential must have the same single frequency dependence. Therefore they can be expressed as the real part of a complex quantity whose time dependence is [math]\displaystyle{ e^{-i\sqrt{\alpha }t} }[/math] where [math]\displaystyle{ \alpha \lt math\gt is the non-dimensional wavenumber and we write }[/math]\bar{W}(\bar{x}, \bar{y},\bar{t})={Re}\left[ w\left( \bar{x},\bar{y}\right) e^{-i\sqrt{ \alpha }\bar{t}}\right] \ [math]\displaystyle{ and }[/math]\;\Phi (\bar{x},\bar{y},\bar{z},\bar{t})= {Re}\left[ \phi \left( \bar{x},\bar{y},\bar{z}\right) e^{-i\sqrt{\alpha }\bar{t}}\right] .</math> In the complex variables the equation of motion of the ice floe ((n-d_ice)) is

[math]\displaystyle{ \beta \nabla ^{4}w+\alpha \gamma w=\sqrt{\alpha }\phi -w. (plate2) }[/math]

From now on we will drop the overbar and assume all variables are non-dimensional.

Equations of Motion for the Water

We require the equation of motion for the water to solve equation (\ref {plate2}). We begin with the non-dimensional equations of potential theory which describe linear surface gravity waves

[math]\displaystyle{ (bvp) \left. \begin{matrix}{rr} \nabla ^{2}\phi =0, & -\infty \lt z\lt 0, \\ {\frac{\partial \phi }{\partial z}=0}, & z\rightarrow -\infty , \\ {\frac{\partial \phi }{\partial z}=}-i\sqrt{\alpha }w, & z\;=\;0,\;\; \mathbf{x}\in \Delta , \\ {\frac{\partial \phi }{\partial z}-}\alpha \phi {=}p, & z\;=\;0,\;\;\mathbf{ x}\notin \Delta , \end{matrix} \right\} (bvp_nond) }[/math]

(Weh_Lait). As before, [math]\displaystyle{ w }[/math] is the displacement of the floe and [math]\displaystyle{ p }[/math] is the pressure at the water surface. The vector [math]\displaystyle{ \mathbf{x=(}x,y) }[/math] is a point on the water surface and [math]\displaystyle{ \Delta }[/math] is the region of the water surface occupied by the floe. The water is assumed infinitely deep. A schematic diagram of this problem is shown in Figure (vibration). \begin{figure}[tbp] \begin{center} \epsfbox{vibration.eps} \end{center} \caption{{The schematic diagram of the boundary value problem and the coordinate system used in the solution.}}

(vibration)

\end{figure}

The boundary value problem ((bvp)) is subject to an incident wave which is imposed through a boundary condition as [math]\displaystyle{ \left| \mathbf{x}\right| \rightarrow \infty }[/math]. This boundary condition, which is called the Sommerfeld radiation condition, is essentially that at large distances the potential consists of a radial outgoing wave (the wave generated by the ice floe motion) and the incident wave. It is expressed mathematically as

[math]\displaystyle{ \lim_{\left| \mathbf{x}\right| \rightarrow \infty }\sqrt{|\mathbf{x}|}\left( \frac{\partial }{\partial |\mathbf{x}|}-i\alpha \right) (\phi -\phi ^{ \mathrm{In}})=0, (summerfield) }[/math]

Weh_Lait. The incident potential (i.e. the incoming wave) [math]\displaystyle{ \phi ^{ \mathrm{In}} }[/math] is

[math]\displaystyle{ \phi ^{\mathrm{In}}(x,y,z)=\frac{A}{\sqrt{\alpha }}e^{i\alpha (x\cos \theta +y\sin \theta )}e^{\alpha z}, (input) }[/math]

where [math]\displaystyle{ A }[/math] is the non-dimensional wave amplitude.

The standard solution method to the linear wave problem is to transform the boundary value problem into an integral equation using a Green function \citep{john1, john2,Sarp_Isa,jgrfloecirc}. Performing such a transformation, the boundary value problem ((bvp)) and ((summerfield)) becomes

[math]\displaystyle{ \phi (\mathbf{x})=\phi ^{i}(\mathbf{x})+\iint_{\Delta }G_{\alpha }(\mathbf{x} ;\mathbf{y})\left( \alpha \phi (\mathbf{x})+i\sqrt{\alpha }w(\mathbf{x} )\right) dS_{\mathbf{y}}. (water) }[/math]

The Green function [math]\displaystyle{ G_{\alpha } }[/math] is

[math]\displaystyle{ G_{\alpha }(\mathbf{x};\mathbf{y)}=\frac{1}{4\pi }\left( \frac{2}{|\mathbf{x} -\mathbf{y}|}-\pi \alpha \left( \mathbf{H_{0}}(\alpha |\mathbf{x}-\mathbf{y} |)+Y_{0}(\alpha |\mathbf{x}-\mathbf{y}|)\right) +2\pi i\alpha J_{0}(\alpha | \mathbf{x}-\mathbf{y}|)\right) , }[/math]

Weh_Lait,jgrfloecirc, where [math]\displaystyle{ J_{0} }[/math] and [math]\displaystyle{ Y_{0} }[/math] are respectively Bessel functions of the first and second kind of order zero, and [math]\displaystyle{ \mathbf{ H_{0}} }[/math] is the Struve function of order zero abr_ste. A solution for water of finite depth could be found by simply using the depth dependent Green function Weh_Lait.

The integral equation ((water)) will be solved using numerical integration. The only difficulty arises from the non-trivial nature of the kernel of the integral equation (the Green function). However, the Green function has no [math]\displaystyle{ z }[/math] dependence due to the shallow draft approximation and depends only on [math]\displaystyle{ |\mathbf{x}-\mathbf{y}|. }[/math] This means that the Green function is one dimensional and the values which are required for a given calculation can be looked up in a previously computed table.

Solving for the Wave Induced Ice Floe Motion

To determine the ice floe motion we must solve equations ((plate2)) and ( (water)) simultaneously. We do this by expanding the floe motion in the free modes of vibration of a thin plate. The major difficulty with this method is that the free modes of vibration can be determined analytically only for very restrictive geometries, e.g. a circular thin plate. Even the free modes of vibration of a square plate with free edges must be determined numerically. This is the reason why the solution of jgrfloecirc was only for a circular floe.

Since the operator [math]\displaystyle{ \nabla ^{4}, }[/math] subject to the free edge boundary conditions, is self adjoint a thin plate must possess a set of modes [math]\displaystyle{ w_{i} }[/math] which satisfy the free boundary conditions and the following eigenvalue equation

[math]\displaystyle{ \nabla ^{4}w_{i}=\lambda _{i}w_{i}. }[/math]

The modes which correspond to different eigenvalues [math]\displaystyle{ \lambda _{i} }[/math] are orthogonal and the eigenvalues are positive and real. While the plate will always have repeated eigenvalues, orthogonal modes can still be found and the modes can be normalized. We therefore assume that the modes are orthonormal, i.e.

[math]\displaystyle{ \iint_{\Delta }w_{i}\left( \mathbf{Q}\right) w_{j}\left( \mathbf{Q}\right) dS_{\mathbf{Q}}=\delta _{ij} }[/math]

where [math]\displaystyle{ \delta _{ij} }[/math] is the Kronecker delta. The eigenvalues [math]\displaystyle{ \lambda _{i} }[/math] have the property that [math]\displaystyle{ \lambda _{i}\rightarrow \infty }[/math] as [math]\displaystyle{ i\rightarrow \infty }[/math] and we order the modes by increasing eigenvalue. These modes can be used to expand any function over the wetted surface of the ice floe [math]\displaystyle{ \Delta }[/math] .

We expand the displacement of the floe in a finite number of modes [math]\displaystyle{ N, }[/math] i.e.

[math]\displaystyle{ w\left( \mathbf{x}\right) =\sum_{i=1}^{N}c_{i}w_{i}\left( \mathbf{x}\right) . (expansion) }[/math]

From the linearity of ((water)) the potential can be written in the following form

[math]\displaystyle{ \phi =\phi _{0}+\sum_{i=1}^{N}c_{i}\phi _{i} (expansionphi) }[/math]

where [math]\displaystyle{ \phi _{0} }[/math] and [math]\displaystyle{ \phi _{i} }[/math] satisfy the integral equations

[math]\displaystyle{ \phi _{0}(\mathbf{x})=\phi ^{\mathrm{In}}(\mathbf{x})+\iint_{\Delta }\alpha G_{\alpha }(\mathbf{x};\mathbf{y})\phi (\mathbf{y})dS_{\mathbf{y}} (phi0) }[/math]

and

[math]\displaystyle{ \phi _{i}(\mathbf{x})=\iint_{\Delta }G_{\alpha }(\mathbf{x};\mathbf{y} )\left( \alpha \phi _{i}(\mathbf{x})+i\sqrt{\alpha }w_{i}(\mathbf{y})\right) dS_{\mathbf{y}}. (phii) }[/math]

The potential [math]\displaystyle{ \phi _{0} }[/math] represents the potential due the incoming wave assuming that the displacement of the ice floe is zero. The potentials [math]\displaystyle{ \phi _{i} }[/math] represent the potential which is generated by the plate vibrating with the [math]\displaystyle{ i }[/math]th mode in the absence of any input wave forcing.

We substitute equations ((expansion)) and ((expansionphi)) into equation ((plate2)) to obtain

[math]\displaystyle{ \beta \sum_{i=1}^{N}\lambda _{i}c_{i}w_{i}-\alpha \gamma \sum_{i=1}^{N}c_{i}w_{i}=i\sqrt{\alpha }\left( \phi _{0}+\sum_{i=1}^{N}c_{i}\phi _{i}\right) -\sum_{i=1}^{N}c_{i}w_{i}. (expanded) }[/math]

To solve equation ((expanded)) we multiply by [math]\displaystyle{ w_{j} }[/math] and integrate over the plate (i.e. we take the inner product with respect to [math]\displaystyle{ w_{j}) }[/math] taking into account the orthogonality of the modes [math]\displaystyle{ w_{i} }[/math], and obtain

[math]\displaystyle{ \beta \lambda _{j}c_{j}+\left( 1-\alpha \gamma \right) c_{j}=\iint_{\Delta }i \sqrt{\alpha }\left( \phi _{0}\left( \mathbf{Q}\right) +\sum_{i=1}^{N}c_{i}\phi _{i}\left( \mathbf{Q}\right) \right) w_{j}\left( \mathbf{Q}\right) dS_{\mathbf{Q}} (final) }[/math]

which is a matrix equation in [math]\displaystyle{ c_{i}. }[/math]

We cannot solve equation ((final)) without determining the modes of vibration of the thin plate [math]\displaystyle{ w_{i} }[/math] (along with the associated eigenvalues [math]\displaystyle{ \lambda _{i}) }[/math] and solving the integral equations ((phi0)) and (\ref {phii}). We use the finite element method to determine the modes of vibration Zienkiewicz and the integral equations ((phi0)) and ( (phii)) are solved by a constant panel method Sarp_Isa. The same set of nodes is used for the finite element method and to define the panels for the integral equation.