Wave Scattering by Submerged Thin Bodies - Revision history

Mike smith: /* Matlab Code */

2009-09-29T22:54:53Z

Matlab Code

2009-09-29T22:54:08Z

References

2009-09-29T22:53:07Z

Eigenfunction Matching

← Older revision		Revision as of 22:53, 29 September 2009
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	== Eigenfunction Matching ==		== Eigenfunction Matching ==
	~~refer~~ to the ~~wikiwaves page etc~~		As an alternative to the the technique outlined on this webpage, [[Eigenfunction Matching for a Submerged Finite Dock]] can be used to determine the potential at any point in the region, as well as the jump in <math>\phi \,</math>

	===Matlab Code ===		===Matlab Code ===

2009-09-29T22:49:29Z

Boundary Element Methods

← Older revision		Revision as of 22:49, 29 September 2009
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	\,\!</math></center>		\,\!</math></center>

	From this we are able to compute the jump in <math>\phi</math>, and the potential around the outer boundary <math>\partial \Omega</math>		From this we are able to compute the jump in <math>\phi \,</math>, and the potential around the outer boundary <math>\partial \Omega</math>

	== Eigenfunction Matching ==		== Eigenfunction Matching ==

2009-09-29T22:49:09Z

Boundary Element Methods

← Older revision		Revision as of 22:49, 29 September 2009
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	Consequently we can represent our system in the following form		Consequently we can represent our system in the following form

			<center><math>
			\begin{bmatrix}
			\tilde{M} & \mathbb{Y}(\mathbb{G}^{(1)} - \mathbb{G}^{(2)}A) \\
			\mathbb{G}^{(3)}\mathbb{W}^{(1)}\mathbb{U} & (\mathbb{G}^{(4)}- \mathbb{G}^{(5)}A - \frac{1}{2}\mathrm{I})
			\end{bmatrix}

			\begin{bmatrix}
			\vec{b} \\
			\vec{\phi}
			\end{bmatrix}

			=

			\begin{bmatrix}
			-\mathbb{Y}\mathbb{G}^{(2)}\vec{f} \\
			-\mathbb{G}^{(5)}\vec{f}
			\end{bmatrix}

			\,\!</math></center>

			From this we are able to compute the jump in <math>\phi</math>, and the potential around the outer boundary <math>\partial \Omega</math>

	== Eigenfunction Matching ==		== Eigenfunction Matching ==

2009-09-29T22:41:45Z

Boundary Element Methods

@@ Line 150: / Line 150: @@
 == Boundary Element Methods ==
-refer to the wikiwaves page etc
+For the purposes of this article, the webpages [[:Category:Boundary Element Method|Boundary Element Method]] and [[Boundary Element Method for a Fixed Body in Finite Depth]] provide an excellent explanation to the Boundary Element method techniques used here.
 == Eigenfunction Matching ==

2009-09-29T22:30:11Z

Hadamard Finite Part Integrals

← Older revision		Revision as of 22:30, 29 September 2009
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	\mathbb{Y} &= \mathbb{U} \ \mathbb{W}^{(1)}\mathbb{W}^{(2)} \\		\mathbb{Y} &= \mathbb{U} \ \mathbb{W}^{(1)}\mathbb{W}^{(2)} \\
	\mathbb{U} &= [U_1(x),U_2(x), ... , U_M(x)]\\		\mathbb{U} &= [U_1(x),U_2(x), ... , U_M(x)]\\
	\mathbb{W}^{(1)}_{nn} &= (1-x_n^2)^{1/2~~}\\~~		\mathbb{W}^{(1)}_{nn} &= (1-x_n^2)^{1/2}
	~~\mathbb{W}^{(2)} &= \mbox{weights matrix for numerical integration~~}
	\end{align}		\end{align}
	</math></center>		</math></center>

	~~The~~ two <math>\mathbb{G}</math> matrices are obtained using boundary element method, which is discussed below.
			Note that <math>\mathbb{W}^{(2)}</math> is a weights matrix for numerically integrating over <math>\Gamma \,</math>, and the two <math>\mathbb{G}</math> matrices are obtained using boundary element method, which is discussed below.

	== Boundary Element Methods ==		== Boundary Element Methods ==

2009-09-28T23:28:02Z

Bibtex

← Older revision		Revision as of 23:28, 28 September 2009
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	The bibtex reference for this webpage is below:		The bibtex reference for this webpage is below:

	@MISC{referenceID,

			<pre>
			@MISC{referenceID,</pre>
	author = {wikiwaves},		author = {wikiwaves},
	title = { {{PAGENAME}} },		title = { {{PAGENAME}} },
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	year = { {{CURRENTYEAR}} },		year = { {{CURRENTYEAR}} },
	url = { {{fullurl:{{PAGENAME}}}} }		url = { {{fullurl:{{PAGENAME}}}} }
			<pre>
	}		}
			</pre>

2009-09-28T23:02:01Z

Hadamard Finite Part Integrals

← Older revision		Revision as of 23:02, 28 September 2009
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	-\frac{1}{2}\sum_{m=0}^{M} b_m (m+1)U_m(x) + \int_{\partial \Omega} \partial_{n^{_{'}}n} G \phi ds^{_{'}}- \int_{\partial \Omega}\partial_n G \partial_{n^{_{'}}} \phi ds^{_{'}} = 0		-\frac{1}{2}\sum_{m=0}^{M} b_m (m+1)U_m(x) + \int_{\partial \Omega} \partial_{n^{_{'}}n} G \phi ds^{_{'}}- \int_{\partial \Omega}\partial_n G \partial_{n^{_{'}}} \phi ds^{_{'}} = 0
	</math></center>		</math></center>


	Multiplying this entire expression by <math>(1-x^2)^{1/2} U_n(x) \,</math> and integrating allows us to obtain the matrix representation		Multiplying this entire expression by <math>(1-x^2)^{1/2} U_n(x) \,</math> and integrating allows us to obtain the matrix representation
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	where		where

			<center><math>
			\begin{align}
			\tilde{M}_{nn} &= -\frac{\pi}{4}n \\
			\mathbb{Y} &= \mathbb{U} \ \mathbb{W}^{(1)}\mathbb{W}^{(2)} \\
			\mathbb{U} &= [U_1(x),U_2(x), ... , U_M(x)]\\
			\mathbb{W}^{(1)}_{nn} &= (1-x_n^2)^{1/2}\\
			\mathbb{W}^{(2)} &= \mbox{weights matrix for numerical integration}
			\end{align}
			</math></center>

	The two <math>\mathbb{G}</math> matrices are obtained using boundary element method, which is discussed below.		The two <math>\mathbb{G}</math> matrices are obtained using boundary element method, which is discussed below.

2009-09-28T22:53:16Z

@@ Line 102: / Line 102: @@
 Note that in our case, <math>\oint \,</math> refers to a Hadamard finite-part integral, as opposed to a contour integral.
 == Boundary Element Methods ==
@@ Line 114: / Line 149: @@
 Article by Yang?
 Linton McIvor?