Panel Methods - Revision history

Adi Kurniawan at 11:14, 27 February 2010

2010-02-27T11:14:17Z

← Older revision		Revision as of 11:14, 27 February 2010
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	With these definitions the Green integral equation may be discretized easily as follows:		With these definitions the Green integral equation may be discretized easily as follows:

	<center><math> \frac{1}{2}\phi_i + \sum_{i=1}^{N} \phi_j \int_{S_j} \mathrm{d}S_X \frac{\partial G ( \vec{X};\vec{\xi}_i )}{\partial n_j}=\sum_{j=1}^{N} V_j \int_{S_j} \mathrm{d}S_X G ( \vec{X};\vec{\xi}_i ), \quad i=1,\cdots,N </math></center>		<center><math> \frac{1}{2}\phi_i + \sum_{j=1}^{N} \phi_j \int_{S_j} \mathrm{d}S_X \frac{\partial G ( \vec{X};\vec{\xi}_i )}{\partial n_j}=\sum_{j=1}^{N} V_j \int_{S_j} \mathrm{d}S_X G ( \vec{X};\vec{\xi}_i ), \quad i=1,\cdots,N </math></center>

	where the integration over the surface of the j-th panel <math>S_j\,</math> is carried out with respect to the <math>\vec{X}\,</math> dummy variable which is allowed to vary over the surface of the j-th panel.		where the integration over the surface of the j-th panel <math>S_j\,</math> is carried out with respect to the <math>\vec{X}\,</math> dummy variable which is allowed to vary over the surface of the j-th panel.

Adi Kurniawan at 10:50, 27 February 2010

2010-02-27T10:50:08Z

← Older revision		Revision as of 10:50, 27 February 2010
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	With these definitions the Green integral equation may be discretized easily as follows:		With these definitions the Green integral equation may be discretized easily as follows:

	<center><math> \frac{1}{2}\phi_i + \sum_{i=1}^{N} \phi_j \int_{S_j} \mathrm{d}S_X \frac{\partial G ~~\left~~( \vec{X};\vec{\xi}_i ~~\right~~)}{\partial n_j}=\sum_{j=1}^{N} V_j \int_{S_j} \mathrm{d}S_X G ~~\left~~( \vec{X};\vec{\xi}_i ~~\right~~), \quad i=1,\cdots,N </math></center>		<center><math> \frac{1}{2}\phi_i + \sum_{i=1}^{N} \phi_j \int_{S_j} \mathrm{d}S_X \frac{\partial G ( \vec{X};\vec{\xi}_i )}{\partial n_j}=\sum_{j=1}^{N} V_j \int_{S_j} \mathrm{d}S_X G ( \vec{X};\vec{\xi}_i ), \quad i=1,\cdots,N </math></center>

	where the integration over the surface of the j-th panel <math>S_j\,</math> is carried out with respect to the <math>\vec{X}\,</math> dummy variable which is allowed to vary over the surface of the j-th panel.		where the integration over the surface of the j-th panel <math>S_j\,</math> is carried out with respect to the <math>\vec{X}\,</math> dummy variable which is allowed to vary over the surface of the j-th panel.

Adi Kurniawan at 10:32, 27 February 2010

2010-02-27T10:32:00Z

← Older revision		Revision as of 10:32, 27 February 2010
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	\| previous chapter = [[Rankine Intergral Equations For Ship Flows]]		\| previous chapter = [[Rankine Intergral Equations For Ship Flows]]
	}}		}}

	{{incomplete pages}}		{{incomplete pages}}

	The principal attributes of the numerical solution of the Green integral equation by panel methods are described below. Most of what follows describes very well the most popular panel methods for the solution of the zero-speed radiation-diffraction problems in the frequency domain. (WAMIT panel method).		The principal attributes of the numerical solution of the Green integral equation by panel methods are described below. Most of what follows describes very well the most popular panel methods for the solution of the zero-speed radiation-diffraction problems in the frequency domain. (WAMIT panel method).

Adi Kurniawan at 10:31, 27 February 2010

2010-02-27T10:31:18Z

← Older revision		Revision as of 10:31, 27 February 2010
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	{{incomplete pages}}		{{incomplete pages}}

	* The principal attributes of the numerical solution of the Green integral equation by panel methods are described below.		The principal attributes of the numerical solution of the Green integral equation by panel methods are described below. Most of what follows describes very well the most popular panel methods for the solution of the zero-speed radiation-diffraction problems in the frequency domain. (WAMIT panel method).

	* Most of ~~what follows describes very well~~ the ~~most popular~~ panel methods for the ~~solution of the zero-speed radiation-diffraction problems~~ in the ~~frequency domain. (WAMIT~~ panel method).		The numerical solution of the Rankine panel methods for the ship flow problem and their extensions, require a stability analysis from first principles. They are not described here but have been developed and implemented in the SWAN panel method.

	* The numerical solution of the ~~Rankine~~ panel methods ~~for~~ the ~~ship flow problem and their extensions, require a stability analysis from first principles. They are~~ not ~~described~~ here ~~but have been developed and implemented in the SWAN panel method~~.		In the zero-speed frequency domain panel methods, the wave Green function is evaluated by specialized very efficient subroutines not discussed further here.

	* In the ~~zero-speed frequency domain panel methods,~~ the ~~wave Green function~~ is ~~evaluated by specialized very efficient subroutines not discussed further here.~~		The first step is to describe the body surface by a collection of quadrilateral panels (facets) arranged in a manner so that the midpoints of adjacent sides coincide. It can be shown that this is always possible!

	* Describe the body surface by a collection of quadrilateral panels ~~(facets) arranged in a manner so~~ that the ~~midpoints of adjacent sides coincide~~. ~~It can be shown that this is always possible!~~		The above arrangement can be shown to leave gaps between panels that attain their maximum magnitude near the vertices. In practice these gaps have been found to lead to no significant errors.

	* The above arrangement can be shown to leave gaps between panels that ~~attain their maximum magnitude near~~ the ~~vertices. In practice these gaps have been found to lead to no significant errors~~.		We assume that over facet <math>i\,</math>, the unknown velocity potential takes a constant value <math>\phi_i,\ i=1,\cdots, N\,</math> where <math>N\,</math> is the total number of facets over <math>S_B\,</math>.

	* Assume that over facet <math>i\,</math>~~, the unknown velocity potential takes a constant value~~ <math>\~~phi_i,\ i=1,\cdots, N\,</math> where <math>N\,</math> is the total number of facets over <math>S_B~~\,</math>.		We define the centroid of the planar quadrilateral denoted as facet <math>i\,</math> by its vector position <math>\vec{X}_i\,</math>

	* Define the centroid of the planar quadrilateral denoted as facet <math>i\,</math> by its vector position <math>\vec{X}_i\,</math>		Let <math>\vec{n}_i\,</math> be the unit normal vector over facet <math>i\,</math>.

	* Let <math>\vec{n}_i\,</math> be the unit normal vector over facet <math>i\,</math>.

	With these definitions the Green integral equation may be discretized easily as follows:		With these definitions the Green integral equation may be discretized easily as follows:

Meylan at 23:37, 16 October 2009

2009-10-16T23:37:17Z

← Older revision		Revision as of 23:37, 16 October 2009
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			{{Ocean Wave Interaction with Ships and Offshore Structures
			\| chapter title = Panel Methods
			\| next chapter = [[Wave Drift Forces]]
			\| previous chapter = [[Rankine Intergral Equations For Ship Flows]]
			}}

			{{incomplete pages}}

	* The principal attributes of the numerical solution of the Green integral equation by panel methods are described below.		* The principal attributes of the numerical solution of the Green integral equation by panel methods are described below.

Adi Kurniawan at 06:24, 24 December 2008

2008-12-24T06:24:59Z

← Older revision		Revision as of 06:24, 24 December 2008
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	With these definitions the Green integral equation may be discretized easily as follows:		With these definitions the Green integral equation may be discretized easily as follows:

	<center><math> \frac{1}{2}\phi_i + \sum_{i=1}^{N} \phi_j \int_{S_j} ~~dS_X~~ \frac{\partial G \left( \vec{X};\vec{\xi}_i \right)}{\partial n_j}=\sum_{j=1}^{N} V_j \int_{S_j} ~~dS_X~~ G \left( \vec{X};\vec{\xi}_i \right), \quad i=1,\cdots,N </math></center>		<center><math> \frac{1}{2}\phi_i + \sum_{i=1}^{N} \phi_j \int_{S_j} \mathrm{d}S_X \frac{\partial G \left( \vec{X};\vec{\xi}_i \right)}{\partial n_j}=\sum_{j=1}^{N} V_j \int_{S_j} \mathrm{d}S_X G \left( \vec{X};\vec{\xi}_i \right), \quad i=1,\cdots,N </math></center>

	where the integration over the surface of the j-th panel <math>S_j\,</math> is carried out with respect to the <math>\vec{X}\,</math> dummy variable which is allowed to vary over the surface of the j-th panel.		where the integration over the surface of the j-th panel <math>S_j\,</math> is carried out with respect to the <math>\vec{X}\,</math> dummy variable which is allowed to vary over the surface of the j-th panel.
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	<center><math> \mathbb{I}: \ \mbox{identity matrix with 1s over the diagonal and zeros elsewhere} \, </math></center>		<center><math> \mathbb{I}: \ \mbox{identity matrix with 1s over the diagonal and zeros elsewhere} \, </math></center>
	<center><math> \mathbb{D} \equiv D_{ij} = \int_{S_j} ~~dS_X~~ \frac{\partial G \left( \vec{X}; \vec{\xi}_i \right)}{\partial n_j} \equiv \ \mbox{dipole influence coefficient matrix}\,</math></center>		<center><math> \mathbb{D} \equiv D_{ij} = \int_{S_j} \mathrm{d}S_X \frac{\partial G \left( \vec{X}; \vec{\xi}_i \right)}{\partial n_j} \equiv \ \mbox{dipole influence coefficient matrix}\,</math></center>
	<center><math> \mathbb{S} \equiv S_{ij} = \int_{S_j} ~~dS_X~~ G \left(\vec{X};\vec{\xi}_i \right) \equiv \ \mbox{source influence coefficient matrix} \, </math></center>		<center><math> \mathbb{S} \equiv S_{ij} = \int_{S_j} \mathrm{d}S_X G \left(\vec{X};\vec{\xi}_i \right) \equiv \ \mbox{source influence coefficient matrix} \, </math></center>

	The solution of the above matrix equation for the unknown vector:		The solution of the above matrix equation for the unknown vector:

Syan077 at 09:06, 19 March 2007

2007-03-19T09:06:15Z

← Older revision		Revision as of 09:06, 19 March 2007
Line 31:		Line 31:
	where:		where:

	<center><math> \mathbb{I}: \ \mbox{identity matrix with ~~1's~~ over the diagonal and zeros elsewhere} \, </math></center>		<center><math> \mathbb{I}: \ \mbox{identity matrix with 1s over the diagonal and zeros elsewhere} \, </math></center>
	<center><math> \mathbb{D} \equiv D_{ij} = \int_{S_j} dS_X \frac{\partial G \left( \vec{X}; \vec{\xi}_i \right)}{\partial n_j} \equiv \ \mbox{dipole influence coefficient matrix}\,</math></center>		<center><math> \mathbb{D} \equiv D_{ij} = \int_{S_j} dS_X \frac{\partial G \left( \vec{X}; \vec{\xi}_i \right)}{\partial n_j} \equiv \ \mbox{dipole influence coefficient matrix}\,</math></center>
	<center><math> \mathbb{S} \equiv S_{ij} = \int_{S_j} dS_X G \left(\vec{X};\vec{\xi}_i \right) \equiv \ \mbox{source influence coefficient matrix} \, </math></center>		<center><math> \mathbb{S} \equiv S_{ij} = \int_{S_j} dS_X G \left(\vec{X};\vec{\xi}_i \right) \equiv \ \mbox{source influence coefficient matrix} \, </math></center>

Syan077 at 09:04, 19 March 2007

2007-03-19T09:04:28Z

New page

* The principal attributes of the numerical solution of the Green integral equation by panel methods are described below.

* Most of what follows describes very well the most popular panel methods for the solution of the zero-speed radiation-diffraction problems in the frequency domain. (WAMIT panel method).

* The numerical solution of the Rankine panel methods for the ship flow problem and their extensions, require a stability analysis from first principles. They are not described here but have been developed and implemented in the SWAN panel method.

* In the zero-speed frequency domain panel methods, the wave Green function is evaluated by specialized very efficient subroutines not discussed further here.

* Describe the body surface by a collection of quadrilateral panels (facets) arranged in a manner so that the midpoints of adjacent sides coincide. It can be shown that this is always possible!

* The above arrangement can be shown to leave gaps between panels that attain their maximum magnitude near the vertices. In practice these gaps have been found to lead to no significant errors.

* Assume that over facet <math>i\,</math>, the unknown velocity potential takes a constant value <math>\phi_i,\ i=1,\cdots, N\,</math> where <math>N\,</math> is the total number of facets over <math>S_B\,</math>.

* Define the centroid of the planar quadrilateral denoted as facet <math>i\,</math> by its vector position <math>\vec{X}_i\,</math>

* Let <math>\vec{n}_i\,</math> be the unit normal vector over facet <math>i\,</math>.

With these definitions the Green integral equation may be discretized easily as follows:

<center><math> \frac{1}{2}\phi_i + \sum_{i=1}^{N} \phi_j \int_{S_j} dS_X \frac{\partial G \left( \vec{X};\vec{\xi}_i \right)}{\partial n_j}=\sum_{j=1}^{N} V_j \int_{S_j} dS_X G \left( \vec{X};\vec{\xi}_i \right), \quad i=1,\cdots,N </math></center>

where the integration over the surface of the j-th panel <math>S_j\,</math> is carried out with respect to the <math>\vec{X}\,</math> dummy variable which is allowed to vary over the surface of the j-th panel.

The location of the i-th vector <math>\vec{\xi}_i\,</math> is fixed and points to the centroid of the i-th panel.

In matrix notation

<center><math> \left[ \frac{1}{2} \mathbb{I} + \mathbb{D} \right] \vec{\phi} = \mathbb{S} \vec{V} \, </math></center>

where:

<center><math> \mathbb{I}: \ \mbox{identity matrix with 1's over the diagonal and zeros elsewhere} \, </math></center>
<center><math> \mathbb{D} \equiv D_{ij} = \int_{S_j} dS_X \frac{\partial G \left( \vec{X}; \vec{\xi}_i \right)}{\partial n_j} \equiv \ \mbox{dipole influence coefficient matrix}\,</math></center>
<center><math> \mathbb{S} \equiv S_{ij} = \int_{S_j} dS_X G \left(\vec{X};\vec{\xi}_i \right) \equiv \ \mbox{source influence coefficient matrix} \, </math></center>

The solution of the above matrix equation for the unknown vector:

<center><math> \vec{\phi} = \left( \phi_1, \cdots, \phi_i, \cdots, \phi_N \right)^T \, </math></center>

In terms of the known vector of normal velocities:

<center><math> \vec{V} = \left( V_1, \cdots, V_i, \cdots, V_N \right)^T \, </math></center>

May be carried out with standard direct or iterative dense matrix solvers.

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]