Eigenfunction Matching Method for Floating Elastic Plates - Revision history

Meylan at 00:05, 17 October 2009

2009-10-17T00:05:01Z

← Older revision		Revision as of 00:05, 17 October 2009
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	{{~~complete~~ pages}}		{{pages to be deleted}}

	==Introduction==		==Introduction==

Meylan at 00:03, 17 October 2009

2009-10-17T00:03:44Z

← Older revision		Revision as of 00:03, 17 October 2009
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	=Introduction=		{{complete pages}}

			==Introduction==

	We show here a solution to the problem of wave propagation under many floating elastic plates of variable properties		We show here a solution to the problem of wave propagation under many floating elastic plates of variable properties
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	the plate boundary conditions are satisfied as auxiliary equations.		the plate boundary conditions are satisfied as auxiliary equations.

	= Equations =		== Equations ==

	We consider the problem of small-amplitude waves which are incident on a set of floating elastic		We consider the problem of small-amplitude waves which are incident on a set of floating elastic
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	shown in Figure~35.		shown in Figure~35.

	=Method of solution=		==Method of solution==

	==Eigenfunction expansion==		===Eigenfunction expansion===

	We will solve the system of equations using an [[:Category:Eigenfunction Matching Method\|Eigenfunction Matching Method]].		We will solve the system of equations using an [[:Category:Eigenfunction Matching Method\|Eigenfunction Matching Method]].
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	the transmission and reflection are not expanded at opposite ends of the plate.		the transmission and reflection are not expanded at opposite ends of the plate.

	===Separation of variables===		====Separation of variables====
	The potential velocity can be written in terms of an infinite series of separated eigenfunctions under		The potential velocity can be written in terms of an infinite series of separated eigenfunctions under
	each elastic plate, of the form		each elastic plate, of the form
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	boundary and by applying the boundary conditions at the edge of each plate.		boundary and by applying the boundary conditions at the edge of each plate.

	==Expressions for the potential velocity==		===Expressions for the potential velocity===

	We now expand the potential under each plate using the separation of variables solution.		We now expand the potential under each plate using the separation of variables solution.

Adi Kurniawan: /* Solving via eigenfunction matching */

2008-12-24T05:41:54Z

Solving via eigenfunction matching

← Older revision		Revision as of 05:41, 24 December 2008
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	\begin{matrix}		\begin{matrix}
	{		{
	{\mathbf M}^{+}_{T_\mu}(m,n) = \int_{-h}^0 e^{-\kappa_\mu(n) (r_\mu-l_\mu )} \frac{\cos \left(k_{\mu}(n) (z+h)\right)}{\cos \left(k_{\mu}(n) h\right)} \cos \left(\frac{m\pi}{h}(z+h)\right) \, dz}, \\		{\mathbf M}^{+}_{T_\mu}(m,n) = \int_{-h}^0 e^{-\kappa_\mu(n) (r_\mu-l_\mu )} \frac{\cos \left(k_{\mu}(n) (z+h)\right)}{\cos \left(k_{\mu}(n) h\right)} \cos \left(\frac{m\pi}{h}(z+h)\right) \, \mathrm{d}z}, \\
	{		{
	{\mathbf M}^{+}_{R_\mu}(m,n) = \int_{-h}^0 \frac{\cos \left(k_{\mu}(n) (z+h)\right)}{\cos \left(k_{\mu}(n) h\right)} \cos \left(\frac{m\pi}{h}(z+h)\right) \, dz },\\		{\mathbf M}^{+}_{R_\mu}(m,n) = \int_{-h}^0 \frac{\cos \left(k_{\mu}(n) (z+h)\right)}{\cos \left(k_{\mu}(n) h\right)} \cos \left(\frac{m\pi}{h}(z+h)\right) \, \mathrm{d}z },\\
	{		{
	{\mathbf M}^{-}_{T_\mu}(m,n) = {\mathbf M}^{+}_{R_\mu}(m,n) }\\		{\mathbf M}^{-}_{T_\mu}(m,n) = {\mathbf M}^{+}_{R_\mu}(m,n) }\\
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	<math>\mathbf{C}</math> is a <math>(M+1)</math> vector which is given by		<math>\mathbf{C}</math> is a <math>(M+1)</math> vector which is given by
	<center><math>		<center><math>
	{\mathbf C}(m)=\int_{-h}^0 \frac{\cos (k_1(0)(z+h))}{\cos (k_1(0)h)} \cos \left(\frac{m\pi}{h}(z+h)\right)\, dz.		{\mathbf C}(m)=\int_{-h}^0 \frac{\cos (k_1(0)(z+h))}{\cos (k_1(0)h)} \cos \left(\frac{m\pi}{h}(z+h)\right)\, \mathrm{d}z.
	</math></center>		</math></center>

Adi Kurniawan: /* Solving via eigenfunction matching */

2008-12-24T05:41:10Z

Solving via eigenfunction matching

← Older revision		Revision as of 05:41, 24 December 2008
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	\begin{matrix}		\begin{matrix}
	{		{
	\int_{-h}^0 \phi_\mu(r_\mu,z)\cos \frac{m\pi}{h}(z+h) \, dz } &=&		\int_{-h}^0 \phi_\mu(r_\mu,z)\cos \frac{m\pi}{h}(z+h) \, \mathrm{d}z } &=&
	{		{
	\int_{-h}^0 \phi_{\mu+1}(l_{\mu+1},z)\cos \frac{m\pi}{h}(z+h) \, dz }\\		\int_{-h}^0 \phi_{\mu+1}(l_{\mu+1},z)\cos \frac{m\pi}{h}(z+h) \, \mathrm{d}z }\\
	{		{
	\int_{-h}^0 \frac{\partial\phi_\mu}{\partial x}(r_\mu,z) \cos \frac{m\pi}{h}(z+h) \, dz } &=&		\int_{-h}^0 \frac{\partial\phi_\mu}{\partial x}(r_\mu,z) \cos \frac{m\pi}{h}(z+h) \, \mathrm{d}z } &=&
	{		{
	\int_{-h}^0 \frac{\partial\phi_{\mu+1}}{\partial x}(l_{\mu+1},z) \cos \frac{m\pi}{h}(z+h) \, dz }		\int_{-h}^0 \frac{\partial\phi_{\mu+1}}{\partial x}(l_{\mu+1},z) \cos \frac{m\pi}{h}(z+h) \, \mathrm{d}z }
	\end{matrix}		\end{matrix}
	</math></center>		</math></center>

Meylan: /* Solving via eigenfunction matching */

2006-12-19T08:45:58Z

Solving via eigenfunction matching

← Older revision		Revision as of 08:45, 19 December 2006
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	Taking inner products leads to the following equations		Taking inner products leads to the following equations
	<center><math>~~(20)~~		<center><math>
	\begin{matrix}		\begin{matrix}
	{		{
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	</math></center>		</math></center>
	where <math>m\in[0,M]</math> and <math>\phi_\mu</math> denotes the potential under the <math>\mu</math>th plate, i.e. the expression		where <math>m\in[0,M]</math> and <math>\phi_\mu</math> denotes the potential under the <math>\mu</math>th plate, i.e. the expression
	for <math>\phi</math> ~~given by Eq.~(19)~~ valid for <math>l_\mu <x<r_\mu</math>.		for <math>\phi</math> valid for <math>l_\mu <x<r_\mu</math>.
	The remaining equations to be solved are given by the two edge conditions satisfied at both		The remaining equations to be solved are given by the two edge conditions satisfied at both
	edges of each plate		edges of each plate
	<center><math>~~(21)~~		<center><math>
	\begin{matrix}		\begin{matrix}
	{		{
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	We will show the explicit form of the linear system of equations which arise		We will show the explicit form of the linear system of equations which arise
	when we solve ~~Eqs.~(20) and (21)~~.		when we solve these equations.
	Let <math>{\mathbf T}_\mu</math> be a column vector given by		Let <math>{\mathbf T}_\mu</math> be a column vector given by
	<math>\left[T_{\mu}(-2), . . ., T_{\mu}(M)\right]^{{\mathbf T}}</math>		<math>\left[T_{\mu}(-2), . . ., T_{\mu}(M)\right]^{{\mathbf T}}</math>
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	The equations which arise from matching at the boundary between the first		The equations which arise from matching at the boundary between the first
	and second plate are		and second plate are
	<center><math>~~(22)~~		<center><math>
	\begin{matrix}		\begin{matrix}
	I{\mathbf C} + {\mathbf M}^{+}_{R_1} {\mathbf R}_1 ={\mathbf M}^{-}_{T_2} {\mathbf T}_2		I{\mathbf C} + {\mathbf M}^{+}_{R_1} {\mathbf R}_1 ={\mathbf M}^{-}_{T_2} {\mathbf T}_2
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	The equations which arise from matching at the boundary of the <math>\mu</math>th and (<math>\mu+1</math>)th plate		The equations which arise from matching at the boundary of the <math>\mu</math>th and (<math>\mu+1</math>)th plate
	boundary (<math>\mu>1</math>) are		boundary (<math>\mu>1</math>) are
	<center><math>~~(23)~~		<center><math>
	\begin{matrix}		\begin{matrix}
	{\mathbf M}^{+}_{T_\mu} {\mathbf T}_\mu +{\mathbf M}^{+}_{R_\mu} {\mathbf R}_\mu		{\mathbf M}^{+}_{T_\mu} {\mathbf T}_\mu +{\mathbf M}^{+}_{R_\mu} {\mathbf R}_\mu
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	</math></center>		</math></center>
	The equations which arise from matching at the (<math>\Lambda-1</math>)th and <math>\Lambda</math>th boundary are		The equations which arise from matching at the (<math>\Lambda-1</math>)th and <math>\Lambda</math>th boundary are
	<center><math>~~(24)~~		<center><math>
	\begin{matrix}		\begin{matrix}
	{\mathbf M}^{+}_{T_{\Lambda-1}} {\mathbf T}_{\Lambda-1}		{\mathbf M}^{+}_{T_{\Lambda-1}} {\mathbf T}_{\Lambda-1}
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	where <math>{\mathbf M}^{+}_{T_\mu}</math>, <math>{\mathbf M}^{+}_{R_\mu}</math>,		where <math>{\mathbf M}^{+}_{T_\mu}</math>, <math>{\mathbf M}^{+}_{R_\mu}</math>,
	<math>{\mathbf M}^{-}_{T_\mu}</math>, and <math>{\mathbf M}^{-}_{R_\mu}</math>are <math>(M+1)</math> by <math>(M+3)</math> matrices given by		<math>{\mathbf M}^{-}_{T_\mu}</math>, and <math>{\mathbf M}^{-}_{R_\mu}</math>are <math>(M+1)</math> by <math>(M+3)</math> matrices given by
	<center><math>~~(25)~~		<center><math>
	\begin{matrix}		\begin{matrix}
	{		{
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	</math></center>		</math></center>
	<math>\mathbf{C}</math> is a <math>(M+1)</math> vector which is given by		<math>\mathbf{C}</math> is a <math>(M+1)</math> vector which is given by
	<center><math>~~(26)~~		<center><math>
	{\mathbf C}(m)=\int_{-h}^0 \frac{\cos (k_1(0)(z+h))}{\cos (k_1(0)h)} \cos \left(\frac{m\pi}{h}(z+h)\right)\, dz.		{\mathbf C}(m)=\int_{-h}^0 \frac{\cos (k_1(0)(z+h))}{\cos (k_1(0)h)} \cos \left(\frac{m\pi}{h}(z+h)\right)\, dz.
	</math></center>		</math></center>

	The integrals in ~~Eqs.~(25) and (26)~~ are each solved analytically. Now, for all but the first and <math>\Lambda</math>th plate, ~~Eq.~(21)~~ becomes		The integrals in the above equation are each solved analytically. Now, for all but the first and <math>\Lambda</math>th plate, the edge equation becomes
	<center><math>		<center><math>
	\begin{matrix}		\begin{matrix}
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	</math></center>		</math></center>

			Now, the matching matrix is a <math>(2M+6)\times(\Lambda-1)</math> by
	~~\noindent~~ Now, the matching matrix is a <math>(2M+6)\times(\Lambda-1)</math> by
	<math>(2M+1)\times(\Lambda -1)</math> matrix given by		<math>(2M+1)\times(\Lambda -1)</math> matrix given by
	<center><math>		<center><math>
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	</math></center>		</math></center>

	~~\noindent~~ the edge matrix is a <math>(2M+6)\times(\Lambda-1)</math> by <math>4(\Lambda-1)</math> matrix given by		the edge matrix is a <math>(2M+6)\times(\Lambda-1)</math> by <math>4(\Lambda-1)</math> matrix given by

	<center><math>		<center><math>
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	</math></center>		</math></center>

	~~\noindent~~ and finally the complete system to be solved is given by		and finally the complete system to be solved is given by
	<center><math>		<center><math>
	\left( \begin{matrix}		\left( \begin{matrix}
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	The final system of equations has size <math>(2M+6)\times (\Lambda - 1)</math> by		The final system of equations has size <math>(2M+6)\times (\Lambda - 1)</math> by
	<math>(2M+6)\times (\Lambda - 1)</math>.		<math>(2M+6)\times (\Lambda - 1)</math>.
	~~The method of~~
	solution we have derived is relatively simple and leads to large systems of equations when we simulate multiple plates. Our aim is to produce code which is simple to develop and which we have a strong degree of confidence is numerically accurate and error free. We do not want to make any kind of wide-spacing approximations since real ice fields always have some small floes which we want to be able to simulate. We have used our method to solve for up to a hundred plates in simulations of wave propagation in the marginal ice zone.

	~~The system of equations has a large~~
	~~number of zero entries, due to the fact that each plate couples~~
	only with its nearest neighbour. It seems likely that a more sophisticated method of solution could be developed, which exploits this structure. We have have been unable to find such a method due to the difficulty of including the free edge conditions.





	[[Category:Floating Elastic Plate]]		[[Category:Floating Elastic Plate]]
	[[Category:Eigenfunction Matching Method]]		[[Category:Eigenfunction Matching Method]]

Meylan: /* Expressions for displacement */

2006-12-19T08:40:42Z

Expressions for displacement

← Older revision		Revision as of 08:40, 19 December 2006
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	<center><math>		<center><math>
	\eta \approx \frac{i}{\omega}\left\{		\eta \approx \frac{i}{\omega}\left\{
	\begin{matrix~~}{ll~~}		\begin{matrix}
	{		{
	Ik_1(0)e^{\kappa_{1}(0)(x-r_1)}\tan{(k_1(0)h)} - } \\		Ik_1(0)e^{\kappa_{1}(0)(x-r_1)}\tan{(k_1(0)h)} - } \\
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	\end{matrix}\right.		\end{matrix}\right.
	</math></center>		</math></center>


	==Solving via eigenfunction matching==		==Solving via eigenfunction matching==

Meylan: /* Expressions for the potential velocity */

2006-12-19T08:40:29Z

Expressions for the potential velocity

← Older revision		Revision as of 08:40, 19 December 2006
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	at <math>M</math> real roots of the dispersion equation.		at <math>M</math> real roots of the dispersion equation.
	The potential <math>\phi</math> can now be expressed as the following sum of eigenfunctions:		The potential <math>\phi</math> can now be expressed as the following sum of eigenfunctions:
	<center><math>~~(19)~~		<center><math>
	\phi \approx \left\{		\phi \approx \left\{
	\begin{matrix}		\begin{matrix}

Meylan: /* Separation of variables */

2006-12-19T08:39:30Z

Separation of variables

← Older revision		Revision as of 08:39, 19 December 2006
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	we obtain the [[Dispersion Relation for a Floating Elastic Plate]]		we obtain the [[Dispersion Relation for a Floating Elastic Plate]]
	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	k_\mu\tan{(k_\mu h)}= & -\frac{\alpha}{\beta_\mu k_\mu^{4} + 1 - \alpha\gamma_\mu}		k_\mu\tan{(k_\mu h)}= & -\frac{\alpha}{\beta_\mu k_\mu^{4} + 1 - \alpha\gamma_\mu}
	~~(18)~~
	\end{matrix}</math></center>		\end{matrix}</math></center>
	Solving for <math>k_\mu</math>~~, this dispersion Eq.~\eqref{DispersionIce}~~ gives a pure imaginary root		Solving for <math>k_\mu</math> gives a pure imaginary root
	with positive imaginary part, two complex roots (two complex conjugate paired roots		with positive imaginary part, two complex roots (two complex conjugate paired roots
	with positive imaginary part in all physical situations), an infinite number of positive real roots		with positive imaginary part in all physical situations), an infinite number of positive real roots
	which approach <math>{n\pi}/{h}</math> as <math>n</math> approaches infinity, and also the negative of all		which approach <math>{n\pi}/{h}</math> as <math>n</math> approaches infinity, and also the negative of all
	these roots [[~~Fox and Squire 1994~~]] . We denote the two complex roots with positive imaginary part		these roots ([[Dispersion Relation for a Floating Elastic Plate]]) . We denote the two complex roots with positive imaginary part
	by <math>k_\mu(-2)</math> and <math>k_\mu(-1)</math>, the purely imaginary		by <math>k_\mu(-2)</math> and <math>k_\mu(-1)</math>, the purely imaginary
	root with positive imaginary part by <math>k_\mu(0)</math> and the real roots with positive imaginary part		root with positive imaginary part by <math>k_\mu(0)</math> and the real roots with positive imaginary part

Meylan: /* Eigenfunction expansion */

2006-12-19T08:38:33Z

Eigenfunction expansion

← Older revision		Revision as of 08:38, 19 December 2006
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	==Eigenfunction expansion==		==Eigenfunction expansion==

	We will solve ~~Eqs.~(13) to (17)~~ using an [[:Category:Eigenfunction Matching Method\|Eigenfunction Matching Method]].		We will solve the system of equations using an [[:Category:Eigenfunction Matching Method\|Eigenfunction Matching Method]].
	The method was developed by [[Fox and Squire 1994]].		The method was developed by [[Fox and Squire 1994]] for the case of a single plate
			as the research is described in [[Two-Dimensional Floating Elastic Plate]].
	We show here how this method can be extended to the case of an arbitrary number of plates.		We show here how this method can be extended to the case of an arbitrary number of plates.
	One of the key features in the eigenfunction expansion method for elastic plates is		One of the key features in the eigenfunction expansion method for elastic plates is
	that extra modes are required in order to solve the higher order boundary		that extra modes are required in order to solve the higher order boundary
	conditions at the plate edges.		conditions at the plate edges.
	~~The first and last plates are semi-infinite and the middle plates are finite.~~
	The potential velocity of the first plate can be expressed as the summation of an incident wave and		The potential velocity of the first plate can be expressed as the summation of an incident wave and
	of reflected waves, one of which is propagating but the rest of		of reflected waves, one of which is propagating but the rest of
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	The potential velocity can be written in terms of an infinite series of separated eigenfunctions under		The potential velocity can be written in terms of an infinite series of separated eigenfunctions under
	each elastic plate, of the form		each elastic plate, of the form
	<~~math></~~math>\phi = e^{\kappa_\mu x} \cos(k_\mu(z+h)).~~<math>~~</math>		<math>\phi = e^{\kappa_\mu x} \cos(k_\mu(z+h)).</math>
	If we apply the boundary conditions given ~~by Eqs.~(14)~~		If we apply the boundary conditions given
	~~and (15)~~
	we obtain the [[Dispersion Relation for a Floating Elastic Plate]]		we obtain the [[Dispersion Relation for a Floating Elastic Plate]]
	<center><math>\begin{matrix}		<center><math>\begin{matrix}

Meylan: /* Equations */

2006-12-19T08:34:06Z

Equations

← Older revision		Revision as of 08:34, 19 December 2006
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	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	\left( \beta_\mu \left(\frac{\partial^2}{\partial x^2} - k^2_y\right)^2		\left( \beta_\mu \left(\frac{\partial^2}{\partial x^2} - k^2_y\right)^2
	- \gamma_\mu\alpha + 1\right)\frac{\partial \phi}{\partial z} - \alpha\phi = 0 \;\;\;\;		- \gamma_\mu\alpha + 1\right)\frac{\partial \phi}{\partial z} - \alpha\phi = 0, \;\;\;\;
	\mbox{ at } z = 0, \;\;\; l_\mu \leq x \leq r_\mu,		\mbox{ at } z = 0, \;\;\; l_\mu \leq x \leq r_\mu,
	\end{matrix}</math></center>		\end{matrix}</math></center>
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	at the edges of the plates are		at the edges of the plates are
	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	\left(\frac{\partial^3}{\partial x^3} - (2 - \nu)k^2_y\frac{\partial}{\partial x}\right) \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; \mbox{ for } x = l_\mu,r_\mu,		\left(\frac{\partial^3}{\partial x^3} - (2 - \nu)k^2_y\frac{\partial}{\partial x}\right) \frac{\partial\phi}{\partial z}= 0, \;\;\;\; \mbox{ at } z = 0, \;\;\; \mbox{ for } x = l_\mu,r_\mu,
	\end{matrix}</math></center>		\end{matrix}</math></center>
	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	\left(\frac{\partial^2}{\partial x^2} - \nu k^2_y\right)\frac{\partial\phi}{\partial z} = 0~~\mbox{ for }~~ \;\;\;\; \mbox{ at } z = 0 \;\;\; \mbox{ for } x = l_\mu,r_\mu.		\left(\frac{\partial^2}{\partial x^2} - \nu k^2_y\right)\frac{\partial\phi}{\partial z} = 0,
			\;\;\;\;\mbox{ at } z = 0, \;\;\; \mbox{ for } x = l_\mu,r_\mu.
	\end{matrix}</math></center>		\end{matrix}</math></center>
	where		where