Three-Dimensional Floating Elastic Plate

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We develop here a theory to solve for a three-dimensional floating elastic plate.

Equations of Motion

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

[math]\displaystyle{ D\nabla ^4 w - \rho _i h \frac{\partial^2 w}{\partial t^2} = p }[/math]

Equation (1) is subject to the free edge boundary conditions for a thin plate

[math]\displaystyle{ \left[ \nabla^2 - (1-\nu) \left(\frac{\partial^2}{\partial s^2} + \kappa(s) \frac{\partial}{\partial n} \right) \right] w = 0, }[/math] [math]\displaystyle{ \left[ \frac{\partial}{\partial n} \nabla^2 +(1-\nu) \frac{\partial}{\partial s} \left( \frac{\partial}{\partial n} \frac{\partial}{\partial s} -\kappa(s) \frac{\partial}{\partial s} \right) \right] w = 0, }[/math]

where [math]\displaystyle{ \nu }[/math] is Poisson's ratio and

[math]\displaystyle{ \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} = \frac{\partial^2}{\partial n^2} + \frac{\partial^2}{\partial s^2} + \kappa(s) \frac{\partial}{\partial n}. }[/math]

Here, [math]\displaystyle{ \kappa(s) }[/math] is the curvature of the boundary, [math]\displaystyle{ \partial \Delta }[/math], as a function of arclength [math]\displaystyle{ s }[/math] along [math]\displaystyle{ \partial \Delta }[/math]; [math]\displaystyle{ \partial/\partial s }[/math] and [math]\displaystyle{ \partial/\partial n }[/math] represent derivatives tangential and normal to the boundary [math]\displaystyle{ \partial \Delta }[/math], respectively (Porter and Porter 2004) 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.\,\,\, (2) }[/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]. In the non-dimensional variables equations (1) and (2) become

[math]\displaystyle{ \beta \nabla^{4}{w}+\gamma \frac{\partial^2 w}{\partial t^2}=\frac{\partial {\Phi}}{\partial {t}}-{w}, \qquad(3)% (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 assume the Frequency Domain Problem with frequency [math]\displaystyle{ \omega }[/math]. This leads to the following equation

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

Equations of Motion for the Water

We require the equation of motion for the water to solve equation ({plate2}). We begin Standard Linear Wave Scattering Problem equations with the boundary condition under the plate modified as appropriate.

[math]\displaystyle{ \left. \begin{matrix} \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\} \qquad(5) }[/math]

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 plate.

The boundary value problem (5) is subject to an incident wave which is imposed through the Sommerfeld Radiation Condition

[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, \qquad(6) }[/math]

where the incident potential [math]\displaystyle{ \phi ^{\mathrm{In}} }[/math] is

[math]\displaystyle{ \phi ^{\mathrm{In}}(x,y,z)=\frac{A}{{\omega }}e^{i\alpha (x\cos \theta +y\sin \theta )}e^{\alpha z}, \qquad (7)(input) }[/math]

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

Solution of the Equations of Motion

There are a number of methods to solve this problem. We will describe a method which generalises the Linear Wave Scattering for a Floating Rigid Body to a plate which has an infinite number of degrees of freedom. Many other methods of solution have been presented, most of which consider some kind of regular plate shape (such as a circle or square). The standard solution method to the linear wave problem is to transform the boundary value problem into an integral equation using a Green function Performing such a transformation, the boundary value problem (5) and (6) become

[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}}. \qquad(8)(water) }[/math]

where [math]\displaystyle{ G_{\alpha } }[/math] is the Free-Surface Green Function

Solving for the Elastic Plate 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 Meylan and Squire 1996 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.