Category:Boundary Element Method - Revision history

Adi Kurniawan: /* Two-Dimensional Equations */

2009-06-24T10:01:10Z

Two-Dimensional Equations

← Older revision		Revision as of 10:01, 24 June 2009
Line 20:		Line 20:
	We apply [http://en.wikipedia.org/w/index.php?title=Green%27s_identities&action=edit Green's second identity]		We apply [http://en.wikipedia.org/w/index.php?title=Green%27s_identities&action=edit Green's second identity]
	to a finite region <math>\Omega</math>. This gives us		to a finite region <math>\Omega</math>. This gives us
	<center><math> \int_\Omega \left( G \nabla^2 \phi - \phi \nabla^2 G\right)\, dV = \oint_{\partial \Omega} \left( G {\partial \phi \over \partial n} - \phi {\partial G \over \partial n}\right)\, dS		<center><math> \int_\Omega \left( G \nabla^2 \phi - \phi \nabla^2 G\right)\, \mathrm{d}V = \oint_{\partial \Omega} \left( G {\partial \phi \over \partial n} - \phi {\partial G \over \partial n}\right)\, \mathrm{d}S
	</math></center>		</math></center>
	We then substitute		We then substitute
Line 38:		Line 38:
	G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi \left( \mathbf{x}		G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi \left( \mathbf{x}
	^{\prime }\right) -G\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi		^{\prime }\right) -G\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi
	_{n}\left( \mathbf{x}^{\prime }\right) \right) d\mathbf{x}^{\prime }		_{n}\left( \mathbf{x}^{\prime }\right) \right) \mathrm{d}\mathbf{x}^{\prime }
	=		=
	\left(		\left(
Line 58:		Line 58:
	G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi \left( \mathbf{x}		G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi \left( \mathbf{x}
	^{\prime }\right) -G\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi		^{\prime }\right) -G\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi
	_{n}\left( \mathbf{x}^{\prime }\right) \right) d\mathbf{x}^{\prime },\;\;		_{n}\left( \mathbf{x}^{\prime }\right) \right) \mathrm{d}\mathbf{x}^{\prime },\;\;
	\mathbf{x}\in \partial \Omega .		\mathbf{x}\in \partial \Omega .
	</math></center>		</math></center>
Line 91:		Line 91:
	<center><math>		<center><math>
	\int_{\mathbf{x}_{i}-h/2}^{\mathbf{x}_{i}+h/2}G\left( \mathbf{x},\mathbf{x}		\int_{\mathbf{x}_{i}-h/2}^{\mathbf{x}_{i}+h/2}G\left( \mathbf{x},\mathbf{x}
	^{\prime }\right) \phi \left( \mathbf{x}^{\prime }\right) d\mathbf{x}		^{\prime }\right) \phi \left( \mathbf{x}^{\prime }\right) \mathrm{d}\mathbf{x}
	^{\prime }\approx \phi \left( \mathbf{x}_{i}\right) \int_{\mathbf{x}		^{\prime }\approx \phi \left( \mathbf{x}_{i}\right) \int_{\mathbf{x}
	_{i}-h/2}^{\mathbf{x}_{i}+h/2}G\left( \mathbf{x},\mathbf{x}^{\prime }\right)		_{i}-h/2}^{\mathbf{x}_{i}+h/2}G\left( \mathbf{x},\mathbf{x}^{\prime }\right)
	d\mathbf{x}^{\prime },		\mathrm{d}\mathbf{x}^{\prime },
	</math></center>		</math></center>
	where <math>\mathbf{x}_{i}</math> is the midpoint of the panel and <math>h</math> is the panel		where <math>\mathbf{x}_{i}</math> is the midpoint of the panel and <math>h</math> is the panel

Meylan: /* Two-Dimensional Equations */

2009-02-20T09:02:50Z

Two-Dimensional Equations

← Older revision		Revision as of 09:02, 20 February 2009
Line 75:		Line 75:
	The outward normal derivative of the potential, <math>\vec{\phi}_{n},</math> and the		The outward normal derivative of the potential, <math>\vec{\phi}_{n},</math> and the
	potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>		potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
	\partial \Omega ~~</math> given by <math>L~~</math>.		\partial \Omega </math>.

	==Numerical Calculation of <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>==		==Numerical Calculation of <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>==

Meylan: /* Two-Dimensional Equations */

2009-02-19T01:17:22Z

Two-Dimensional Equations

← Older revision		Revision as of 01:17, 19 February 2009
Line 42:		Line 42:
	\left(		\left(
	\begin{matrix}		\begin{matrix}
	0, \,\,\,x\notin U \cup \partial U, \\		0, \,\,\,x\notin \Omega \cup \partial \Omega, \\
	\phi(\mathbf{x})/2,\,\,\,\mathbf{x} \in \partial U, \\		\phi(\mathbf{x})/2,\,\,\,\mathbf{x} \in \partial \Omega, \\
	\phi(\mathbf{x}),\,\,\,\mathbf{x} \in U,		\phi(\mathbf{x}),\,\,\,\mathbf{x} \in \Omega,
	\end{matrix}		\end{matrix}
	\right.		\right.
Line 52:		Line 52:
	coordinate).		coordinate).
	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 U </math>		and its outward normal derivative on the boundary <math>\partial \Omega </math>
	<center><math>		<center><math>
	\left. \frac{1}{2}\phi \left( \mathbf{x}\right) \right\|_{\mathbf{x}\in\partial U}		\left. \frac{1}{2}\phi \left( \mathbf{x}\right) \right\|_{\mathbf{x}\in\partial \Omega}
	=\oint_{\partial \Omega }\left(		=\oint_{\partial \Omega }\left(
	G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi \left( \mathbf{x}		G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi \left( \mathbf{x}
Line 75:		Line 75:
	The outward normal derivative of the potential, <math>\vec{\phi}_{n},</math> and the		The outward normal derivative of the potential, <math>\vec{\phi}_{n},</math> and the
	potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>		potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
	\partial U </math> given by <math>L</math>.		\partial \Omega </math> given by <math>L</math>.

	==Numerical Calculation of <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>==		==Numerical Calculation of <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>==

Meylan: /* Two-Dimensional Equations */

2009-02-19T01:11:36Z

Two-Dimensional Equations

← Older revision		Revision as of 01:11, 19 February 2009
Line 19:		Line 19:

	We apply [http://en.wikipedia.org/w/index.php?title=Green%27s_identities&action=edit Green's second identity]		We apply [http://en.wikipedia.org/w/index.php?title=Green%27s_identities&action=edit Green's second identity]
	to a finite region ~~''U''~~. This gives us		to a finite region <math>\Omega</math>. This gives us
	<center><math> \~~int_U~~ \left( G \nabla^2 \phi - \phi \nabla^2 G\right)\, dV = \oint_{\partial U} \left( G {\partial \phi \over \partial n} - \phi {\partial G \over \partial n}\right)\, dS		<center><math> \int_\Omega \left( G \nabla^2 \phi - \phi \nabla^2 G\right)\, dV = \oint_{\partial \Omega} \left( G {\partial \phi \over \partial n} - \phi {\partial G \over \partial n}\right)\, dS
	</math></center>		</math></center>
	We then substitute		We then substitute

Meylan: /* Two-Dimensional Equations */

2008-08-18T03:34:46Z

Two-Dimensional Equations

← Older revision		Revision as of 03:34, 18 August 2008
Line 19:		Line 19:

	We apply [http://en.wikipedia.org/w/index.php?title=Green%27s_identities&action=edit Green's second identity]		We apply [http://en.wikipedia.org/w/index.php?title=Green%27s_identities&action=edit Green's second identity]
	to a finite region ''U'' ~~containing the wetted~~		to a finite region ''U''. This gives us
	~~body surface <math>\partial\Omega</math>, the free surface <math>z=0</math>,~~
	~~the bottom surface <math>z=-h</math> and two vertical boundaries~~. This gives us
	<center><math> \int_U \left( G \nabla^2 \phi - \phi \nabla^2 G\right)\, dV = \oint_{\partial U} \left( G {\partial \phi \over \partial n} - \phi {\partial G \over \partial n}\right)\, dS		<center><math> \int_U \left( G \nabla^2 \phi - \phi \nabla^2 G\right)\, dV = \oint_{\partial U} \left( G {\partial \phi \over \partial n} - \phi {\partial G \over \partial n}\right)\, dS
	</math></center>		</math></center>

Meylan: /* Two-Dimensional Equations */

2008-08-18T03:34:17Z

Two-Dimensional Equations

@@ Line 18: / Line 18: @@
 = Two-Dimensional Equations =
-We begin with the [[Standard Linear Wave Scattering Problem]]
+We apply [http://en.wikipedia.org/w/index.php?title=Green%27s_identities&action=edit Green's second identity]
 to a finite region ''U'' containing the wetted
 body surface <math>\partial\Omega</math>, the free surface <math>z=0</math>,

Meylan: /* Matlab Code */

2008-08-18T03:26:25Z

Matlab Code

← Older revision		Revision as of 03:26, 18 August 2008
Line 122:		Line 122:
	= Matlab Code =		= Matlab Code =

			{{boundary element code}}

	[[Category:Pages with Matlab Code]]		[[Category:Pages with Matlab Code]]
	[[Category:Numerical Methods]]		[[Category:Numerical Methods]]

Meylan: /* Numerical Calculation of $\mathbf{G}$ and $\mathbf{G}_{n}$ */

2008-08-18T03:26:09Z

Numerical Calculation of <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>

← Older revision		Revision as of 03:26, 18 August 2008
Line 118:		Line 118:

	It is well known that, for higher accuracy, a higher order boundary element method works		It is well known that, for higher accuracy, a higher order boundary element method works
	better.		better.

	~~{{boundary element code}}~~

	= Matlab Code =		= Matlab Code =

Meylan: /* Matlab Code */

2008-08-18T03:25:05Z

Matlab Code

← Older revision		Revision as of 03:25, 18 August 2008
Line 119:		Line 119:
	It is well known that, for higher accuracy, a higher order boundary element method works		It is well known that, for higher accuracy, a higher order boundary element method works
	better.		better.

			{{boundary element code}}

	= Matlab Code =		= Matlab Code =

	~~A program to calculate the matrics <math>G</math> and <math>G_{n}</math> can be found here~~
	~~[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/bem_constant_panel.m bem_constant_panel.m]~~


	[[Category:Pages with Matlab Code]]		[[Category:Pages with Matlab Code]]
	[[Category:Numerical Methods]]		[[Category:Numerical Methods]]

Meylan: /* Introduction */

2008-08-18T02:50:35Z

Introduction

← Older revision		Revision as of 02:50, 18 August 2008
Line 14:		Line 14:
	is used so that the problem reduces to solving only on the wetted body surface.		is used so that the problem reduces to solving only on the wetted body surface.

	The theory for solving a wave scattering problem is described in [[Boundary Element Method for a Rigid Body]]		The theory for solving a wave scattering problem is described in [[Boundary Element Method for a Rigid Body in Finite Depth]]

	= Two-Dimensional Equations =		= Two-Dimensional Equations =

Category:Boundary Element Method - Revision history

Adi Kurniawan: /* Two-Dimensional Equations */

Meylan: /* Two-Dimensional Equations */

Meylan: /* Two-Dimensional Equations */

Meylan: /* Two-Dimensional Equations */

Meylan: /* Two-Dimensional Equations */

Meylan: /* Two-Dimensional Equations */

Meylan: /* Matlab Code */

Meylan: /* Numerical Calculation of \mathbf{G} and \mathbf{G}_{n} */

Meylan: /* Matlab Code */

Meylan: /* Introduction */

Meylan: /* Numerical Calculation of $\mathbf{G}$ and $\mathbf{G}_{n}$ */