Floating Elastic Plate on Variable Bottom Topography - Revision history

Meylan at 09:22, 20 October 2009

2009-10-20T09:22:23Z

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			{{pages to be deleted}}

	=Introduction=		=Introduction=

Meylan: /* Numerical Solution Method */

2008-05-01T20:02:15Z

Numerical Solution Method

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	boundaries it is given by an integral equation. We must solve both Laplace's		boundaries it is given by an integral equation. We must solve both Laplace's
	equation and the integral equations numerically. We will solve Laplace's		equation and the integral equations numerically. We will solve Laplace's
	equation by the [[Boundary Element Method]] and the integral equations by		equation by the [[:Category:Boundary Element Method\|Boundary Element Method]] and the integral equations by
	numerical integration. However, the same discretisation of the boundary will		numerical integration. However, the same discretisation of the boundary will
	be used for both numerical solutions.		be used for both numerical solutions.

	We begin by applying the [[Boundary Element Method]].		We begin by applying the [[:Category:Boundary Element Method\|Boundary Element Method]].
	This gives us the following equation relating the potential		This gives us the following equation relating the potential
	and its outward normal derivative on the boundary <math>\partial \Omega </math>		and its outward normal derivative on the boundary <math>\partial \Omega </math>

Meylan: /* Problem Formulation */

2006-08-18T06:54:17Z

Problem Formulation

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	\nabla ^{2}\phi =0, \\		\nabla ^{2}\phi =0, \\
	-\rho \left( g\phi _{n}-\omega ^{2}\,\phi \right) =\left\{		-\rho \left( g\phi _{n}-\omega ^{2}\,\phi \right) =\left\{
	\begin{matrix~~}{c~~}		\begin{matrix}
	0,\;x\notin \left[ -L,L\right] ,\;z=0, \\		0,\;x\notin \left[ -L,L\right] ,\;z=0, \\
	D\,\frac{\partial ^{4}\phi _{n}}{\partial x^{4}}-\rho ^{\prime }a\,\omega		D\,\frac{\partial ^{4}\phi _{n}}{\partial x^{4}}-\rho ^{\prime }a\,\omega

Meylan: /* Numerical Solution Method */

2006-08-17T10:23:14Z

Numerical Solution Method

Meylan: /* Solution in the Semi-infinite Domains */

2006-08-17T10:19:58Z

Solution in the Semi-infinite Domains

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	</math></center>		</math></center>
	The normal derivative of the potential on the boundary of <math>\Omega ^{-}</math> and <math>		The normal derivative of the potential on the boundary of <math>\Omega ^{-}</math> and <math>
	\Omega <math> </math>\left( x=-l\right) </math> is calculated using equation (14) and		\Omega </math> <math>\left( x=-l\right) </math> is calculated using equation (14) and
	we obtain,		we obtain,
	<center><math>		<center><math>

Meylan: /* Non-dimensionalisation */

2006-08-17T10:19:05Z

Non-dimensionalisation

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	<center><math>		<center><math>
	\bar{x}=\frac{x}{H_{1}},\;\bar{z}=\frac{z}{H_{1}},\;\bar{t}=t\sqrt{\frac{g		\bar{x}=\frac{x}{H_{1}},\;\bar{z}=\frac{z}{H_{1}},\;\bar{t}=t\sqrt{\frac{g
	}{H_{1}}},\;=and= \;\bar{\phi}=\frac{1}{H_{1}\sqrt{H_{1}g}}\,\phi .		}{H_{1}}},\;\mathrm{and} \;\bar{\phi}=\frac{1}{H_{1}\sqrt{H_{1}g}}\,\phi .
	</math></center>		</math></center>
	Applying this non-dimensionalisation to equation (4)		Applying this non-dimensionalisation to equation (4)
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	<center><math>		<center><math>
	\beta =\frac{D}{\rho gH_{1}^{4}},\;\gamma =\frac{\rho ^{\prime }a}{\rho		\beta =\frac{D}{\rho gH_{1}^{4}},\;\gamma =\frac{\rho ^{\prime }a}{\rho
	H_{1}},\;=and= \;\nu =\frac{\omega ^{2}H_{1}}{g}. (10)		H_{1}},\;\mathrm{and} \;\nu =\frac{\omega ^{2}H_{1}}{g}. (10)
	</math></center>		</math></center>
	We will refer to <math>\beta </math> as the stiffness, <math>\gamma </math> as the mass and <math>\nu </math>		We will refer to <math>\beta </math> as the stiffness, <math>\gamma </math> as the mass and <math>\nu </math>

Meylan: /* Solution in the Semi-infinite Domains */

2006-08-17T10:18:13Z

Solution in the Semi-infinite Domains

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	</math></center>		</math></center>
	The evanescent eigenvalues <math>k_{m}^{\left( 1\right) }</math> are the positive real		The evanescent eigenvalues <math>k_{m}^{\left( 1\right) }</math> are the positive real
	solutions of the ~~dispersion equation~~		solutions of the [[Dispersion Relation for a Free Surface]]
	<center><math>		<center><math>
	-k_{m}^{\left( 1\right) }\,\tan \left( k_{m}^{\left( 1\right) }H_{j}\right)		-k_{m}^{\left( 1\right) }\,\tan \left( k_{m}^{\left( 1\right) }H_{j}\right)

Meylan: /* Radiation Boundary Conditions */

2006-08-17T10:17:03Z

Radiation Boundary Conditions

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	where <math>R</math> and <math>T</math> are the reflection and transmission coefficients		where <math>R</math> and <math>T</math> are the reflection and transmission coefficients
	respectively and <math>k_{t}^{\left( j\right) }\,\left( j=1,2\right) </math> are the		respectively and <math>k_{t}^{\left( j\right) }\,\left( j=1,2\right) </math> are the
	positive real solutions of the ~~following equation~~		positive real solutions of the [[Dispersion Relation for a Free Surface]]
	<center><math>		<center><math>
	gk_{t}^{\left( j\right) }\tanh \left( k_{t}^{\left( j\right) }\,H_{j}\right)		gk_{t}^{\left( j\right) }\tanh \left( k_{t}^{\left( j\right) }\,H_{j}\right)

Meylan: /* Introduction */

2006-08-17T10:14:41Z

Introduction

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	=Introduction=		=Introduction=

	A solution method for the linear wave forcing of a two		We present here the solution of [[Wang and Meylan 2004]].
	dimensional floating plate on water of variable depth ~~will be~~ derived from		The solution method for the linear wave forcing of a two
			dimensional floating plate on water of variable depth is derived from
	[[Liu and Liggett 1982]], [[Hazard and Lenoir 1993]] and [[Meylan and Squire 1994]]. The method is based on		[[Liu and Liggett 1982]], [[Hazard and Lenoir 1993]] and [[Meylan and Squire 1994]]. The method is based on
	dividing the water domain into two semi-infinite domains and a finite		dividing the water domain into two semi-infinite domains and a finite

Meylan: /* Summary */

2006-08-17T10:03:53Z

Summary

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	necessarily as strong.		necessarily as strong.

	~~=Summary=~~

	~~We have presented a solution for the linear wave forcing of a floating thin~~
	~~plate on water of variable depth. The solution method was based on reducing~~
	~~the problem to a finite domain which contained both the region of variable~~
	~~water depth and the floating thin plate. In this finite region the outward~~
	~~normal derivative of the potential around the boundary was expressed as a~~
	~~function of the potential. This was accomplished by using integral operators~~
	~~for the boundary under the plate and the radiating boundaries. The integral~~
	~~operator for the plate was calculated using a Green function as described in~~
	~~[[jgrfloe1d]]. The integral operators for the radiation boundary~~
	~~conditions were calculated by solving Laplace's equation in the~~
	~~semi-infinite outer domains using separation of variables as described in~~
	~~[[Liu82, Hazard]]. Laplace's equation in the finite domain was solved~~
	~~using the boundary element method.~~

	~~The results showed that, for certain parameter regimes, there was~~
	~~significant difference between the absolute value of the reflection~~
	~~coefficient for the variable depth and constant depth profiles. Furthermore,~~
	~~the region of influence of the variable depth was increased by the presence~~
	~~of the plate due to the increased wavelength under the plate. Finally, there~~
	~~was a slight increase in the bending of the plate due to the presence of the~~
	~~variable depth profile.~~

	~~\bibliographystyle{IEEE}~~
	~~\bibliography{mike,others}~~
	~~\pagebreak~~

	=Tables=		=Tables=

← Older revision		Revision as of 10:23, 17 August 2006
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	boundaries it is given by an integral equation. We must solve both Laplace's		boundaries it is given by an integral equation. We must solve both Laplace's
	equation and the integral equations numerically. We will solve Laplace's		equation and the integral equations numerically. We will solve Laplace's
	equation by the ~~boundary element method~~ and the integral equations by		equation by the [[Boundary Element Method]] and the integral equations by
	numerical integration. However, the same discretisation of the boundary will		numerical integration. However, the same discretisation of the boundary will
	be used for both numerical solutions.		be used for both numerical solutions.

	We begin by applying the ~~boundary element method to equation (\ref~~		We begin by applying the [[Boundary Element Method]].
	~~{finitedomain}).~~ This gives us the following equation relating the potential		This gives us the following equation relating the potential
	and its outward normal derivative on the boundary <math>\partial \Omega </math>		and its outward normal derivative on the boundary <math>\partial \Omega </math>
	<center><math>		<center><math>
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	because of the simple structure of <math>G</math>, can be calculated exactly.		because of the simple structure of <math>G</math>, can be calculated exactly.

	==Numerical Calculation of <math>\mathbf{Q}_{1}</math> and <math>\mathbf{Q}_{2} </math>}==		==Numerical Calculation of <math>\mathbf{Q}_{1}</math> and <math>\mathbf{Q}_{2} </math>==

	We will discuss the numerical approximation of the operator <math>\mathbf{Q}_{1}</math>.		We will discuss the numerical approximation of the operator <math>\mathbf{Q}_{1}</math>.